Displaying 1-10 of 48 results found.
a(0) = 1 and a(n) = A180000(n)*a(floor(n/2))^2 for n > 0.
+20
2
1, 1, 1, 1, 2, 2, 3, 3, 48, 16, 40, 40, 270, 270, 945, 63, 129024, 129024, 64512, 64512, 2016000, 96000, 528000, 528000, 144342000, 28868400, 187644600, 20849400, 1787836050, 1787836050, 59594535, 59594535, 3999321544458240, 121191561953280, 1030128276602880
COMMENTS
lcm(1,2,..,n) = (n!*a(n)) / ((n/2)!*a(n/2))^2.
lcm(1,2,..,n)*a(n) is a divisor of n! and n!/(lcm(1,2,..,n)*a(n)) is a square.
PROG
(Sage)
if n == 0 : return 1
Swinging factorial, a(n) = 2^(n-(n mod 2))*Product_{k=1..n} k^((-1)^(k+1)).
+10
147
1, 1, 2, 6, 6, 30, 20, 140, 70, 630, 252, 2772, 924, 12012, 3432, 51480, 12870, 218790, 48620, 923780, 184756, 3879876, 705432, 16224936, 2704156, 67603900, 10400600, 280816200, 40116600, 1163381400, 155117520, 4808643120, 601080390, 19835652870, 2333606220
COMMENTS
a(n) is the number of 'swinging orbitals' which are enumerated by the trinomial n over [floor(n/2), n mod 2, floor(n/2)].
Similar to but different from A001405(n) = binomial(n, floor(n/2)), a(n) = lcm( A001405(n-1), A001405(n)) (for n>0). Exactly p consecutive multiples of p follow the least positive multiple of p if p is an odd prime. Compare with the similar property of A100071. - Peter Luschny, Aug 27 2012 a(n) is the number of vertices of the polytope resulting from the intersection of an n-hypercube with the hyperplane perpendicular to and bisecting one of its long diagonals. - Didier Guillet, Jun 11 2018 [Edited by Peter Munn, Dec 06 2022]
FORMULA
a(n) = n!/floor(n/2)!^2. [Essentially the original name.]
a(0) = 1, a(n) = n^(n mod 2)*(4/n)^(n+1 mod 2)*a(n-1) for n>=1.
O.g.f.: a(n) = SeriesCoeff_{n}((1+z/(1-4*z^2))/sqrt(1-4*z^2)).
P.g.f.: a(n) = PolyCoeff_{n}((1+z^2)^n+n*z*(1+z^2)^(n-1)).
a(2*n) = binomial(2*n,n); a(2*n+1) = (2*n+1)*binomial(2*n,n). Central terms of triangle A211226. - Peter Bala, Apr 10 2012 D-finite with recurrence: n*a(n) + (n-2)*a(n-1) + 4*(-2*n+3)*a(n-2) + 4*(-n+1)*a(n-3) + 16*(n-3)*a(n-4) = 0. - Alexander R. Povolotsky, Aug 17 2012 E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+1)/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012 Central column of the coefficients of the swinging polynomials A162246. - Peter Luschny, Oct 22 2013 a(n) = hypergeometric([-n,-n-1,1/2],[-n-2,1],2)*2^(n-1)*(n+2). - Peter Luschny, Sep 22 2014 a(n) = 4^floor(n/2)*hypergeometric([-floor(n/2), (-1)^n/2], [1], 1). - Peter Luschny, May 19 2015 Sum_{n>=0} (-1)^n/a(n) = 4/3 - 4*Pi/(9*sqrt(3)). - Amiram Eldar, Mar 10 2022
EXAMPLE
a(10) = 10!/5!^2 = trinomial(10,[5,0,5]);
a(11) = 11!/5!^2 = trinomial(11,[5,1,5]).
MAPLE
SeriesCoeff := proc(s, n) series(s(w, n), w, n+2);
convert(%, polynom); coeff(%, w, n) end;
a1 := proc(n) local k;
2^(n-(n mod 2))*mul(k^((-1)^(k+1)), k=1..n) end:
a2 := proc(n) option remember;
`if`(n=0, 1, n^irem(n, 2)*(4/n)^irem(n+1, 2)*a2(n-1)) end;
a3 := n -> n!/iquo(n, 2)!^2;
g4 := z -> BesselI(0, 2*z)*(1+z);
a4 := n -> n!*SeriesCoeff(g4, n);
g5 := z -> (1+z/(1-4*z^2))/sqrt(1-4*z^2);
a5 := n -> SeriesCoeff(g5, n);
g6 := (z, n) -> (1+z^2)^n+n*z*(1+z^2)^(n-1);
a6 := n -> SeriesCoeff(g6, n);
a7 := n -> combinat[multinomial](n, floor(n/2), n mod 2, floor(n/2));
h := n -> binomial(n, floor(n/2)); # A001405a8 := n -> ilcm(h(n-1), h(n));
F := [a1, a2, a3, a4, a5, a6, a7, a8];
for a in F do seq(a(i), i=0..32) od;
MATHEMATICA
f[n_] := 2^(n - Mod[n, 2])*Product[k^((-1)^(k + 1)), {k, n}]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 02 2010 *) f[n_] := If[OddQ@n, n*Binomial[n - 1, (n - 1)/2], Binomial[n, n/2]]; Array[f, 33, 0] (* Robert G. Wilson v, Aug 10 2010 *) sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; (* or, twice faster: *) sf[n_] := n!/Quotient[n, 2]!^2; Table[sf[n], {n, 0, 32}] (* Jean-François Alcover, Jul 26 2013, updated Feb 11 2015 *)
PROG
(Magma) [(Factorial(n)/(Factorial(Floor(n/2)))^2): n in [0..40]]; // Vincenzo Librandi, Sep 11 2011 (Sage)
r, n = 1, 0
while True:
yield r
n += 1
r *= 4/n if is_even(n) else n
Numerators in expansion of 1/sqrt(1-x).
(Formerly M2508 N0992)
+10
78
1, 1, 3, 5, 35, 63, 231, 429, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 9694845, 300540195, 583401555, 2268783825, 4418157975, 34461632205, 67282234305, 263012370465, 514589420475, 8061900920775, 15801325804719
COMMENTS
Also numerator of e(n-1,n-1) (see Maple line).
Leading coefficient of normalized Legendre polynomial.
Common denominator of expansions of powers of x in terms of Legendre polynomials P_n(x).
Also the numerator of binomial(2*n,n)/2^n. - T. D. Noe, Nov 29 2005 This sequence gives the numerators of the Maclaurin series of the Lorentz factor (see Wikipedia link) of 1/sqrt(1-b^2) = dt/dtau where b=u/c is the velocity in terms of the speed of light c, u is the velocity as observed in the reference frame where time t is measured and tau is the proper time. - Stephen Crowley, Apr 03 2007 Truncations of rational expressions like those given by the numerator operator are artifacts in integer formulas and have many disadvantages. A pure integer formula follows. Let n$ denote the swinging factorial and sigma(n) = number of '1's in the base-2 representation of floor(n/2). Then a(n) = (2*n)$ / sigma(2*n) = A056040(2*n) / A060632(2*n+1). Simply said: this sequence is the odd part of the swinging factorial at even indices. - Peter Luschny, Aug 01 2009 The convolution of sequence binomial(2*n,n)/4^n with itself is the constant sequence with all terms = 1.
a(n) equals the denominator of Hypergeometric2F1[1/2, n, 1 + n, -1] (see Mathematica code below). - John M. Campbell, Jul 04 2011 a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2-2*x+2)^n dx. - Leonid Bedratyuk, Nov 17 2012 a(n) = numerator of the mean value of cos(x)^(2*n) from x = 0 to 2*Pi. - Jean-François Alcover, Mar 21 2013 Constant terms for normalized Legendre polynomials. - Tom Copeland, Feb 04 2016 By analytic continuation to the entire complex plane there exist regularized values for divergent sums:
a(n)/ A060818(n) = (-2)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)). Sum_{k>=0} (-1)^k*a(k)/ A060818(k) = 1/sqrt(3). Sum_{k>=0} (-1)^(k+1)*a(k)/ A060818(k) = -1/sqrt(3). a(n)/ A046161(n) = (-1)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)). Sum_{k>=0} (-1)^k*a(k)/ A046161(k) = 1/sqrt(2). Sum_{k>=0} (-1)^(k+1)*a(k)/ A046161(k) = -1/sqrt(2). (End) a(n) = numerator of (1/Pi)*Integral_{x=-oo..+oo} 1/(x^2+1)^n dx. (n=1 is the Cauchy distribution.) - Harry Garst, May 26 2017
Let R(n, d) = (Product_{j prime to d} Pochhammer(j / d, n)) / n!. Then the numerators of R(n, 2) give this sequence and the denominators are A046161. For d = 3 see A273194/ A344402. - Peter Luschny, May 20 2021 Using WolframAlpha, it appears a(n) gives the numerator in the residues of f(z) = 2z choose z at odd negative half integers. E.g., the residues of f(z) at z = -1/2, -3/2, -5/2 are 1/(2*Pi), 1/(16*Pi), and 3/(256*Pi) respectively. - Nicholas Juricic, Mar 31 2022 a(n) is the numerator of (1/Pi) * Integral_{x=-oo..+oo} sech(x)^(2*n+1) dx. The corresponding denominator is A046161. - Mohammed Yaseen, Jul 29 2023 a(n) is the numerator of (1/Pi) * Integral_{x=0..Pi/2} sin(x)^(2*n) dx. The corresponding denominator is A101926(n). - Mohammed Yaseen, Sep 19 2023
REFERENCES
P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 372.
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 2nd ed. New York: Wiley, 1968; Chap. III, Eq. 4.1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equation 6:14:6 at page 51.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 102.
FORMULA
a(n) = numerator( binomial(2*n,n)/4^n ) (cf. A046161). a(n) = denominator of (2^n/binomial(2*n,n)). - Artur Jasinski, Nov 26 2011 a(n) = numerator(L(n)), with rational L(n):=binomial(2*n,n)/2^n. L(n) is the leading coefficient of the Legendre polynomial P_n(x).
L(n) = (2*n-1)!!/n! with the double factorials (2*n-1)!! = A001147(n), n >= 0. Numerator in (1-2t)^(-1/2) = 1 + t + (3/2)t^2 + (5/2)t^3 + (35/8)t^4 + (63/8)t^5 + (231/16)t^6 + (429/16)t^7 + ... = 1 + t + 3*t^2/2! + 15*t^3/3! + 105*t^4/4! + 945*t^5/5! + ... = e.g.f. for double factorials A001147 (cf. A094638). - Tom Copeland, Dec 04 2013 a(n)/ A061549(n) = (-1/4)^n*sqrt(Pi)/(Gamma(1/2 - n)*Gamma(1 + n)). Sum_{k>=0} a(k)/ A061549(k) = 2/sqrt(3). Sum_{k>=0} (-1)^k*a(k)/ A061549(k) = 2/sqrt(5). Sum_{k>=0} (-1)^(k+1)*a(k)/ A061549(k) = -2/sqrt(5). a(n)/ A123854(n) = (-1/2)^n*sqrt(Pi)/(gamma(1/2 - n)*gamma(1 + n)). Sum_{k>=0} a(k)/ A123854(k) = sqrt(2). Sum_{k>=0} (-1)^k*a(k)/ A123854(k) = sqrt(2/3). Sum_{k>=0} (-1)^(k+1)*a(k)/ A123854(k) = -sqrt(2/3). (End)
EXAMPLE
1, 1, 3/2, 5/2, 35/8, 63/8, 231/16, 429/16, 6435/128, 12155/128, 46189/256, ...
binomial(2*n,n)/4^n => 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, 6435/32768, ...
MAPLE
e := proc(l, m) local k; add(2^(k-2*m)*binomial(2*m-2*k, m-k)*binomial(m+k, m)*binomial(k, l), k=l..m); end;
swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i, i=convert(iquo(n, 2), base, 2))):
a := n -> swing(2*n)/sigma(2*n); # (End)
MATHEMATICA
Numerator[ CoefficientList[ Series[1/Sqrt[(1 - x)], {x, 0, 25}], x]]
Table[Denominator[Hypergeometric2F1[1/2, n, 1 + n, -1]], {n, 0, 34}] (* John M. Campbell, Jul 04 2011 *) Numerator[Table[(-2)^n*Sqrt[Pi]/(Gamma[1/2 - n]*Gamma[1 + n]), {n, 0, 20}]] (* Ralf Steiner, Apr 07 2017 *) Numerator[Table[Binomial[2n, n]/2^n, {n, 0, 25}]] (* Vaclav Kotesovec, Apr 07 2017 *) Table[Numerator@LegendreP[2 n, 0]*(-1)^n, {n, 0, 25}] (* Andres Cicuttin, Jan 22 2018 *) A = {1}; Do[A = Append[A, 2^IntegerExponent[n, 2]*(2*n - 1)*A[[n]]/n], {n, 1, 25}]; Print[A] (* John Lawrence, Jul 17 2020 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( pollegendre(n), n) * 2^valuation((n\2*2)!, 2))};
(PARI) a(n)=binomial(2*n, n)>>hammingweight(n); \\ Gleb Koloskov, Sep 26 2021 @CachedFunction
def swing(n):
if n == 0: return 1
return swing(n-1)*n if is_odd(n) else 4*swing(n-1)/n
(Magma)
A001790:= func< n | Numerator((n+1)*Catalan(n)/4^n) >;
CROSSREFS
Cf. A060818 (denominator of binomial(2*n,n)/2^n), A061549 (denominators). Cf. A161198 (triangle of coefficients for (1-x)^((-1-2*n)/2)). Cf. A163590 (odd part of the swinging factorial). First column and diagonal 1 of triangle A100258.
Numerators in expansion of (1 - x)^(-3/2).
(Formerly M2986 N1207)
+10
50
1, 3, 15, 35, 315, 693, 3003, 6435, 109395, 230945, 969969, 2028117, 16900975, 35102025, 145422675, 300540195, 9917826435, 20419054425, 83945001525, 172308161025, 1412926920405, 2893136075115, 11835556670925
COMMENTS
a(n) is the denominator of the integral from 0 to Pi of (sin(x))^(2*n+1). - James R. Buddenhagen, Aug 17 2008 a(n) is the denominator of (2n)!!/(2*n + 1)!! = 2^(2*n)*n!*n!/(2*n + 1)! (see Andersson). - N. J. A. Sloane, Jun 27 2011 a(n) = (2*n + 1)* A001790(n). A046161(n)/a(n) = 1, 2/3, 8/15, 16/35, 128/315, 256/693, ... is binomial transform of Madhava-Gregory-Leibniz series for Pi/4 (i.e., 1 - 1/3 + 1/5 - 1/7 + ... ). See A173384 and A173396. - Paul Curtz, Feb 21 2010 a(n) is the denominator of Integral_{x=-oo..oo} sech(x)^(2*n+2) dx. The corresponding numerator is A101926(n). - Mohammed Yaseen, Jul 25 2023
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, equation 6:14:9 at page 51.
LINKS
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
FORMULA
a(n) is the numerator of (2*n + 1)*binomial(2*n,n)/(4^n).
(1 - x)^(-3/2) = Sum_{n>=0} ((2*n + 1)*binomial(2*n,n)/(4^n)*x^n)
(End)
Truncations of rational expressions like those given by the numerator or denominator operators are artifacts in integer formulas and have many disadvantages. A pure integer formula follows. Let n$ denote the swinging factorial and sigma(n) = number of '1's in the base-2 representation of floor(n/2). Then a(n) = (2*n+1)$ / sigma(2*n+1) = A056040(2*n+1) / A060632(2*n+2). Simply said: This sequence gives the odd part of the swinging factorial at odd indices. - Peter Luschny, Aug 01 2009 a(n) = denominator(Pi*binomial(n, -1/2)). - Peter Luschny, Dec 06 2024
MAPLE
swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i, i= convert(iquo(n, 2), base, 2))):
a := n -> swing(2*n+1)/sigma(2*n+1); # Peter Luschny, Aug 01 2009 A001803 := proc(n) (2*n+1)*binomial(2*n, n)/4^n ; numer(%) ; end proc: # R. J. Mathar, Jul 06 2011
a := n -> denom(Pi*binomial(n, -1/2)): seq(a(n), n = 0..22); # Peter Luschny, Dec 06 2024
MATHEMATICA
Numerator/@CoefficientList[Series[(1-x)^(-3/2), {x, 0, 25}], x] (* Harvey P. Dale, Feb 19 2011 *) Table[Denominator[Beta[1, n + 1, 1/2]], {n, 0, 22}] (* Gerry Martens, Nov 13 2016 *)
PROG
(PARI) a(n) = numerator((2*n+1)*binomial(2*n, n)/(4^n)); \\ Altug Alkan, Sep 06 2018 (Julia)
CROSSREFS
The denominator is given in A046161. Cf. A002596 (numerators in expansion of (1-x)^(1/2)). Cf. A161198 (triangle related to the series expansions of (1-x)^((-1-2*n)/2)). (End)
GCD of consecutive central binomial coefficients: a(n) = gcd( A001405(n+1), A001405(n)).
+10
35
1, 1, 1, 3, 2, 10, 5, 35, 14, 126, 42, 462, 132, 1716, 429, 6435, 1430, 24310, 4862, 92378, 16796, 352716, 58786, 1352078, 208012, 5200300, 742900, 20058300, 2674440, 77558760, 9694845, 300540195, 35357670, 1166803110, 129644790, 4537567650
COMMENTS
The numbers can be seen as a generalization of the Catalan numbers, extending A000984(n)/(n+1) to A056040(n)/(floor(n/2)+1). They can also be seen as composing the aerated Catalan numbers A126120 with the aerated complementary Catalan numbers A138364. (Thus the name 'extended Catalan numbers' might be apt for this sequence.) - Peter Luschny, May 03 2011 a(n) is the number of lattice paths from (0,0) to (n,0) that do not go below the x-axis and consist of steps U=(1,1), D=(1,-1) and maximally one step H=(1,0). - Alois P. Heinz, Apr 17 2013 a(n) can be computed with ballot numbers without multiplications or divisions, see Maple program. - Peter Luschny, Feb 23 2019
FORMULA
G.f.: (4*x^2+x-1+(1-x)*sqrt(1-4*x^2))/(2*sqrt(1-4*x^2)*x^2). E.g.f.: (1+1/x)*BesselI(1, 2*x). - Vladeta Jovovic, Jan 19 2004 Recurrence: a(0) = 1 and a(n) = a(n-1)*n^[n odd]*(4/(n+2))^[n even] for n > 0.
Asymptotic formula: Let [n even] = 1 if n is even, 0 otherwise. Let N := n+1+[n even]. Then a(n) ~ 2^N /((n+1)^[n even]*sqrt(Pi*(2*N+1))).
Integral representation: a(n) = (1/(2*Pi))*Int_{x=0..4}(x^(2*n-1)* ((4-x)^2/x)^cos(Pi*n))^(1/4) (End)
E.g.f.: U(0) where U(k)= 1 + x/(1 - x/(x + (k+1)*(k+2)//U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 19 2012 D-finite with recurrence: (n+2)*a(n) - n*a(n-1) + 4*(-2*n+1)*a(n-2) + 4*(n-1)*a(n-3) + 16*(n-3)*a(n-4) = 0.
D-finite with recurrence: -(n+2)*(n^2-5)*a(n) + 4*(-2*n-1)*a(n-1) + 4*(n-1)*(n^2+2*n-4)*a(n-2) = 0. (End)
Sum_{n>=0} 1/a(n) = 8/3 + 8*Pi/(9*sqrt(3)). - Amiram Eldar, Aug 20 2022
EXAMPLE
This GCD equals A001405(n) for the smaller odd number gcd(C(12,6), C(11,5)) = gcd(924,462) = 462 = C(11,5).
MAPLE
A057977_ogf := proc(z) b := z -> (z-1)/(2*z^2);
(2 + b(z))/sqrt(1-4*z^2) - b(z) end:
seq(coeff(series( A057977_ogf(z), z, n+3), z, n), n = 0..35); *(4/(n+2))^modp(n+1, 2));
A057977_int := proc(n) int((x^(2*n-1)*((4-x)^2/x)^cos(Pi*n))^(1/4), x=0..4)/(2*Pi); round(evalf(%)) end:
A057977 := n -> (n!/iquo(n, 2)!^2) / (iquo(n, 2)+1):
b := proc(p, q) option remember; local S;
if p = 0 and q = 0 then return 1 fi;
if p < 0 or p > q then return 0 fi;
S := b(p-2, q) + b(p, q-2);
if type(q, odd) then S := S + b(p-1, q-1) fi;
S end:
PROG
(PARI) a(n)=if(n<0, 0, (n+n%2)!/(n\2+1)!/(n\2+n%2)!/(1+n%2))
(Sage)
x, n = 1, 1
while True:
yield x
m = n if is_odd(n) else 4/(n+2)
x *= m
n += 1
1, 1, 2, 6, 12, 60, 120, 840, 1680, 15120, 30240, 332640, 665280, 8648640, 17297280, 259459200, 518918400, 8821612800, 17643225600, 335221286400, 670442572800, 14079294028800, 28158588057600, 647647525324800, 1295295050649600
COMMENTS
Product of the largest parts in the partitions of n+1 into exactly two parts, n > 0. - Wesley Ivan Hurt, Jan 26 2013 (Clarified on Apr 20 2016)
FORMULA
a(n) = sqrt(n!*n$) where n$ denotes the swinging factorial ( A056040). a(n) = 2^n Gamma((n+1+(n mod 2))/2)/sqrt(Pi). (End)
E.g.f.: E(0) where E(k) = 1 + x/(1 - x/(x + (k+1)/E(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 20 2012 G.f.: G(0) where G(k) = 1 + x*(2*k+1)/(1 - 2*x/(2*x + 1/G(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Nov 18 2012 Conjecture: a(n) +2*a(n-1) -2*n*a(n-2) +4*(-n+2)*a(n-3) = 0. - R. J. Mathar, Nov 26 2012 a(n) = n!/(n-floor((n+1)/2))!.
a(n) = Product_{i = ceiling(n/2)..(n-1)} i. [Note: empty product = 1]
a(n) = P( n, floor((n+1)/2) ), where P(n,k) are the number of k-permutations of n objects. (End)
a(n) = n$*floor(n/2)! where n$ denotes the swinging factorial ( A056040). - Peter Luschny, Oct 28 2013 Sum_{n>=0} 1/a(n) = 1 + (3/2)*exp(1/4)*sqrt(Pi)*erf(1/2).
Sum_{n>=0} (-1)^n/a(n) = 1 - (1/2)*exp(1/4)*sqrt(Pi)*erf(1/2). (End)
EXAMPLE
a(3) = 6 since 3+1 = 4 has two partitions into two parts, (3,1) and (2,2), and the product of the largest parts is 6. - Wesley Ivan Hurt, Jan 26 2013 (Clarified on Apr 20 2016)
MAPLE
Method 1) a:=n->n!/floor(n/2)!; seq(a(k), k=0..40); # Wesley Ivan Hurt, Jun 03 2013 Method 2) with(combinat, numbperm); seq(numbperm(k, floor((k+1)/2)), k = 0..40); # Wesley Ivan Hurt, Jun 06 2013
PROG
(Magma) [Factorial(n)/(Factorial(Floor(n/2))): n in [0..30]]; // Vincenzo Librandi, Sep 13 2011 (Sage)
def a(n): return rising_factorial(ceil(n/2), floor(n/2))
(Python)
from sympy import rf
Extended Motzkin numbers, Sum_{k>=0} C(n,k)*C(k), where C(k) is the extended Catalan number A057977(k).
+10
9
1, 2, 4, 10, 25, 66, 177, 484, 1339, 3742, 10538, 29866, 85087, 243478, 699324, 2015082, 5822619, 16865718, 48958404, 142390542, 414837699, 1210439958, 3536809521, 10347314544, 30306977757, 88861597426, 260798283502, 766092871654, 2252240916665
COMMENTS
a(n) = Sum_{k=0..n} binomial(n,k)* A057977(k). For comparison: Thus one might simply say: The extended Motzkin numbers are the binomial sum of the extended Catalan numbers. Moreover: The Catalan numbers aerated with 0's at odd positions ( A126120) are the inverse binomial transform of the Motzkin numbers ( A001006). The complementary Catalan numbers ( A001700) aerated with 0's at even positions ( A138364) are the inverse binomial transform of the complementary Motzkin numbers ( A005717). The extended Catalan numbers ( A057977 = A126120 + A138364) are the inverse binomial transform of the extended Motzkin numbers ( A189912). David Scambler observed that [1, a(n-1)] for n >= 1 count the Dyck paths of semilength n which satisfy the condition "number of peaks <= number of returns + number of hills". - Peter Luschny, Oct 22 2012
FORMULA
a(n) = Sum_{k=0..n} n!/(((n-k)!*floor(k/2)!^2)*(floor(k/2)+1)).
Recurrence: (n+2)*(n^2 + 2*n - 5)*a(n) = (2*n^3 + 7*n^2 - 14*n - 7)*a(n-1) + 3*(n-1)*(n^2 + 4*n - 2)*a(n-2). - Vaclav Kotesovec, Mar 20 2014
MAPLE
add(n!/(((n-k)!*iquo(k, 2)!^2)*(iquo(k, 2)+1)), k=0..n) end:
M := proc(n) option remember; `if`(n<2, 1, (3*(n-1)*M(n-2)+(2*n+1)*M(n-1))/(n+2)) end:
MATHEMATICA
A057977[n_] := n!/(Quotient[n, 2]!^2*(Quotient[n, 2] + 1)); a[n_] := Sum[Binomial[n, k]* A057977[k], {k, 0, n}]; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, May 21 2013, after Peter Luschny *)
Table[Sum[n!/(((n-k)!*Floor[k/2]!^2)*(Floor[k/2]+1)), {k, 0, n}], {n, 0, 30}] (* G. C. Greubel, Jan 24 2017 *)
PROG
(Sage)
@CachedFunction
def M(n): return (3*(n-1)*M(n-2)+(2*n+1)*M(n-1))/(n+2) if n>1 else 1
A189912 = lambda n: n*M(n-1) + M(n)
(PARI) a(n) = sum(k=0, n, binomial(n, k)*k!/( (k\2)!^2 * (k\2+1)) );
Extended Catalan triangle read by rows.
+10
8
1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 2, 8, 3, 4, 1, 10, 5, 15, 4, 5, 1, 5, 30, 9, 24, 5, 6, 1, 35, 14, 63, 14, 35, 6, 7, 1, 14, 112, 28, 112, 20, 48, 7, 8, 1, 126, 42, 252, 48, 180, 27, 63, 8, 9, 1, 42, 420, 90, 480, 75, 270, 35, 80, 9, 10, 1, 462, 132, 990, 165, 825, 110, 385, 44, 99, 10, 11, 1
COMMENTS
Let S(n,k) denote the coefficients of the positive powers of the Laurent polynomials C_n(x) = (x+1/x)^(n-1)*(x-1/x)*(x+1/x+n) (if n>0) and C_0(x) = 0.
Then T(n,k) = S(n+1,k+1) for n>=0, k>=0.
The classical Catalan triangle A053121(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is odd. The complementary Catalan triangle A189230(n,k) can be recovered from this triangle by setting T(n,k) = 0 if n-k is even. T(n,0) are the extended Catalan numbers A057977(n).
FORMULA
Recurrence: If k>n or k<0 then T(n,k) = 0 else if n=k then T(n,k) = 1; otherwise T(n,k) = T(n-1,k-1) + ((n-k) mod 2)*T(n-1,k) + T(n-1,k+1).
S(n,k) = (k/n)* A162246(n,k) for n>0 where S(n,k) are the coefficients from the definition provided the triangle A162246 is indexed in Laurent style by the recurrence: if abs(k) > n then A162246(n,k) = 0 else if n = k then A162246(n,k) = 1 and otherwise A162246(n,k) = A162246(n-1,k-1)+ modp(n-k,2) * A162246(n-1,k) + A162246(n-1,k+1).
EXAMPLE
The Laurent polynomials:
C(0,x) = 0
C(1,x) = x - 1/x
C(2,x) = x^2 + x - 1/x - 1/x^2
C(3,x) = x^3 + 2 x^2 + x - 1/x - 2/x^2 -1/x^3
Triangle T(n,k) = S(n+1,k+1) starts
[0] 1,
[1] 1, 1,
[2] 1, 2, 1,
[3] 3, 2, 3, 1,
[4] 2, 8, 3, 4, 1,
[5] 10, 5, 15, 4, 5, 1,
[6] 5, 30, 9, 24, 5, 6, 1,
[7] 35, 14, 63, 14, 35, 6, 7, 1,
[0],[1],[2],[3],[4],[5],[6],[7]
MAPLE
A189231_poly := (n, x)-> `if`(n=0, 0, (x+1/x)^(n-2)*(x-1/x)*(x+1/x+n-1)):
seq(print([n], seq(coeff(expand( A189231_poly(n, x)), x, k), k=1..n)), n=1..9); A189231 := proc(n, k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1, A189231(n-1, k-1)+modp(n-k, 2)* A189231(n-1, k)+ A189231(n-1, k+1))) end:
seq(print(seq( A189231(n, k), k=0..n)), n=0..9);
MATHEMATICA
t[n_, k_] /; (k > n || k < 0) = 0; t[n_, n_] = 1; t[n_, k_] := t[n, k] = t[n-1, k-1] + Mod[n-k, 2]*t[n-1, k] + t[n-1, k+1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2013 *)
Swinging polynomials, coefficients read by rows.
+10
7
1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 3, 6, 3, 3, 1, 1, 4, 4, 12, 6, 12, 4, 4, 1, 1, 5, 5, 20, 10, 30, 10, 20, 5, 5, 1, 1, 6, 6, 30, 15, 60, 20, 60, 15, 30, 6, 6, 1, 1, 7, 7, 42, 21, 105, 35, 140, 35, 105, 21, 42, 7, 7, 1
COMMENTS
Let p(n,x) = (1+x^2)^n+n*x*(1+x^2)^(n-1), then T(n,k) are the coefficients of these polynomials, read by rows, n = 0,1,...
The central numbers of the rows, i.e., the coefficients of x^n of p(n,x), are the swinging factorial numbers A056040(n). Row sums: sum_{k=0..2n} T(n,k) = A001792(n). sum_{k=0..2n} isodd(n+k)T(n,k) = 2^n(isodd(n)+(n/2)isodd(n+1))
= 0, 2, 4, 8, 32, 32, 192, 128, 1024, 512, 5120, ...
sum_{k=0..2n} iseven(n+k)T(n,k) = 2^n(isodd(n)(n/2)+isodd(n+1))
= 1, 1, 4, 12, 16, 80, 64, 448, 256, 2304, 1024, ...
FORMULA
T(n,k) = n!/((n-ceiling(k/2))!*floor(k/2)!).
EXAMPLE
The central coefficients are marked by [].
[1]
1,[1],1
1,2,[2],2,1
1,3,3,[6],3,3,1
1,4,4,12,[6],12,4,4,1
1,5,5,20,10,[30],10,20,5,5,1
1,6,6,30,15,60,[20],60,15,30,6,6,1
1,7,7,42,21,105,35,[140],35,105,21,42,7,7,1
p(0,x) = 1
p(1,x) = x^2+x+1
p(2,x) = x^4+2x^3+2x^2+2x+1
p(3,x) = x^6+3x^5+3x^4+6x^3+3x^2+3x+1
p(4,x) = x^8+4x^7+4x^6+12x^5+6x^4+12x^3+4x^2+4x+1
p(5,x) = x^10+5x^9+5x^8+20x^7+10x^6+30x^5+10x^4+20x^3+5x^2+5x+1
MAPLE
p := (n, x) -> (1+x^2)^n+n*x*(1+x^2)^(n-1):
seq(print(seq(coeff(expand(p(n, x)), x, i), i=0..2*n)), n=0..7);
T := (n, k) -> n!/((n-ceil(k/2))!*floor(k/2)!);
seq(print(seq(T(n, k), k=0..2*n)), n=0..7);
MATHEMATICA
t[n_, k_] := If[EvenQ[k], Binomial[n, k/2], Binomial[n, (k-1)/2]*(n-(k-1)/2)]; Table[t[n, k], {n, 0, 7}, {k, 0, 2*n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
Swinging Wilson quotients ((p-1)$ +(-1)^floor((p+2)/2))/p, p prime. Here '$' denotes the swinging factorial function ( A056040).
+10
7
1, 1, 1, 3, 23, 71, 757, 2559, 30671, 1383331, 5003791, 245273927, 3362110459, 12517624987, 175179377183, 9356953451851, 509614686432899, 1938763632210843, 107752663194272623
EXAMPLE
The 5th prime is 11, (11-1)$ = 252, the remainder term is (-1)^floor((11+2)/2)=1. So the quotient (252+1)/11 = 23 is the 5th member of the sequence.
MAPLE
swing := proc(n) option remember; if n = 0 then 1 elif irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
WQ := proc(f, r, n) map(p->(f(p-1)+r(p))/p, select(isprime, [$1..n])) end:
A163210 := n -> WQ(swing, p->(-1)^iquo(p+2, 2), n);
MATHEMATICA
sf[n_] := n!/Quotient[n, 2]!^2; a[n_] := (p = Prime[n]; (sf[p - 1] + (-1)^Floor[(p + 2)/2])/p); Table[a[n], {n, 1, 19}] (* Jean-François Alcover, Jun 28 2013 *) a[p_] := (Binomial[p-1, (p-1)/2] - (-1)^((p-1)/2)) / p
Join[{1, 1}, a[Prime[Range[3, 20]]]] (* Peter Luschny, May 13 2017 *)
PROG
(PARI) a(n, p=prime(n)) = ((p-1)!/((p-1)\2)!^2 - (-1)^(p\2))/p \\ David A. Corneth, May 13 2017
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