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Expansion of exp(x)/(1 - x - x^2/2!), where a(n) = 1 + n*a(n-1) + C(n,2)*a(n-2).
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1, 2, 6, 25, 137, 936, 7672, 73361, 801705, 9856342, 134640146, 2023140417, 33163934641, 588936102860, 11263023492372, 230783643185881, 5044101110058737, 117136294344278346, 2880200768035996990
FORMULA
a(n) ~ n!*exp(sqrt(3)-1)*((1+sqrt(3))/2)^(n+1)/sqrt(3) . - Vaclav Kotesovec, Oct 20 2012
EXAMPLE
E.g.f.: exp(x)/(1 - x - x^2/2!) = 1 + 2*x + 6*x^2/2! + 25*x^3/3! + 137*x^4/4! + 936*x^5/5! + 7672*x^6/6! +... + a(n)*x^n/n! +...
where a(n) = 1 + n*a(n-1) + n*(n-1)*a(n-2)/2.
MATHEMATICA
CoefficientList[Series[E^x/(1-x-x^2/2), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)
PROG
(PARI) a(n)=n!*polcoeff(exp(x+x*O(x^n))/(1-x-x^2/2! +x*O(x^n)), n)
a(1)=0, a(2)=1; for n>2, a(n) = C(n,2)*(1+a(n-2)).
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0, 1, 3, 12, 40, 195, 861, 5488, 31032, 247005, 1706815, 16302396, 133131648, 1483518127, 13978823145, 178022175360, 1901119947856, 27237392830233, 325091511083547, 5175104637744460, 68269217327545080, 1195449171318970491
COMMENTS
a(n) is also the maximum number of ways to place node pairs in an area formed by n 1 X 1 squares. - Theodore M. Mishura, Mar 20 2015
FORMULA
a(n) = Sum_{k=1..floor(n/2)} 2^k*Pochhammer(-n/2,k)*Pochhammer(1/2-n/2,k). - Theodore M. Mishura, Mar 16 2015 a(n) ~ n! * (exp(sqrt(2)) + (-1)^n * exp(-sqrt(2))) / 2^(n/2+1). - Vaclav Kotesovec, Mar 20 2015
MAPLE
seq(simplify(hypergeom([1, 1-n/2, 3/2-n/2], [], 2))*(n-1)*n/2, n=1..22); # Mark van Hoeij, May 12 2013
MATHEMATICA
nxt[{n_, a_, b_}]:={n+1, b, Binomial[n+1, 2](a+1)}; Transpose[NestList[nxt, {2, 0, 1}, 30]][[2]] (* Harvey P. Dale, Oct 12 2014 *)
PROG
(Magma) I:=[0, 1]; [n le 2 select I[n] else Binomial(n, 2)*(1+Self(n-2)): n in [1..35]]; // Vincenzo Librandi, Mar 17 2015
AUTHOR
Allan L. Edmonds (edmonds(AT)indiana.edu), Feb 13 2007
Expansion of exp(x)/(1 - x^3/3!), where a(n) = 1 + binomial(n,3)*a(n-3).
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2
1, 1, 1, 2, 5, 11, 41, 176, 617, 3445, 21121, 101806, 757901, 6040607, 37057385, 344844956, 3382739921, 25199021801, 281393484097, 3277874983450, 28726884853141, 374253333849011, 5047927474513001, 50875313074912712
EXAMPLE
E.g.f.: exp(x)/(1 - x^3/3!) = 1 + x + 1*x^2/2! + 2*x^3/3! + 5*x^4/4! + 11*x^5/5! + 41*x^6/6! + ... + a(n)*x^n/n! + ...
where a(n) = 1 + n*(n-1)*(n-2)*a(n-3)/3!.
MAPLE
restart: G(x):=2*exp(-x)/(x^3/3!+1): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(-f[n-1], x) od: x:=0: seq(f[n]/2, n=0..23); # Zerinvary Lajos, Apr 03 2009
PROG
(PARI) a(n)=n!*polcoeff(exp(x+x*O(x^n))/(1-x^3/3! +x*O(x^n)), n)
Expansion of exp(x)/(1 - x^4/4!), where a(n) = 1 + C(n,4)*a(n-4).
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2
1, 1, 1, 1, 2, 6, 16, 36, 141, 757, 3361, 11881, 69796, 541256, 3364362, 16217566, 127028721, 1288189281, 10294947721, 62859285817, 615454153246, 7709812846786, 75307542579116, 556618975909536, 6539815832391997
EXAMPLE
E.g.f.: exp(x)/(1 - x^4/4!) = 1 + x + 1*x^2/2! + 1*x^3/3! + 2*x^4/4! + 6*x^5/5! + 16*x^6/6! +... + a(n)*x^n/n! +...
where a(n) = 1 + n*(n-1)*(n-2)*(n-3)*a(n-4)/4!.
MAPLE
G(x):=exp(x)/(1-x^4/4!): f[0]:=G(x): for n from 1 to 26 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..24); # Zerinvary Lajos, Apr 03 2009
PROG
(PARI) a(n)=n!*polcoeff(exp(x+x*O(x^n))/(1-x^4/4! +x*O(x^n)), n)
(PARI) /* Recurrence: */ a(n)=if(n<0, 0, if(n<4, 1, 1 + n*(n-1)*(n-2)*(n-3)*a(n-4)/4!))
Exponential Riordan array [1/(1-x^2/2), x].
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1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 6, 0, 6, 0, 1, 0, 30, 0, 10, 0, 1, 90, 0, 90, 0, 15, 0, 1, 0, 630, 0, 210, 0, 21, 0, 1, 2520, 0, 2520, 0, 420, 0, 28, 0, 1, 0, 22680, 0, 7560, 0, 756, 0, 36, 0, 1, 113400, 0, 113400, 0, 18900, 0, 1260, 0, 45, 0, 1
FORMULA
Number triangle T(n,k) = [k<=n](n!/k!)*(1/2^((n-k)/2))*(1+(-1)^(n-k))/2.
EXAMPLE
Triangle begins
1,
0, 1,
1, 0, 1,
0, 3, 0, 1,
6, 0, 6, 0, 1,
0, 30, 0, 10, 0, 1,
90, 0, 90, 0, 15, 0, 1,
0, 630, 0, 210, 0, 21, 0, 1
MATHEMATICA
(* The function RiordanArray is defined in A256893. *)
Triangle T(n,k) read by rows: number of k-lists (ordered k-sets) of disjoint 2-subsets of an n-set, n>1, 0<k<=floor(n/2).
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1, 3, 6, 6, 10, 30, 15, 90, 90, 21, 210, 630, 28, 420, 2520, 2520, 36, 756, 7560, 22680, 45, 1260, 18900, 113400, 113400, 55, 1980, 41580, 415800, 1247400, 66, 2970, 83160, 1247400, 7484400, 7484400, 78, 4290, 154440, 3243240, 32432400, 97297200
COMMENTS
T(n,k) is also the number of involutions (unary operators) on S_n, i.e., endomorphisms U with 2k non-invariant elements such that U^2 is the identity mapping. The extension to n=1 is a(1)=0. - Stanislav Sykora, Nov 03 2016
FORMULA
E.g.f.: y*x^2*exp(x)/(2-y*x^2).
T(n,k) = Product_{m=1..floor(n/2)} binomial(n-2*m,2) = n!/(2^k*(n-2*k)!).
EXAMPLE
For n = 4 we have 12 lists: 6 1-lists: [{1,2}], [{1,3}], [{1,4}], [{2,3}], [{2,4}], [{3,4}] and 6 2-lists: [{1,2},{3,4}], [{3,4},{1,2}], [{1,3},{2,4}], [{2,4},{1,3}], [{1,4},{2,3}] and [{2,3},{1,4}].
MATHEMATICA
Table[n!/(2^k (n - 2 k)!), {n, 2, 13}, {k, Floor[n/2]}] // Flatten (* Michael De Vlieger, Nov 04 2016 *)
PROG
(PARI) nmax=100; a=vector(floor(nmax^2/4)); idx=0;
for(n=2, nmax, for(k=1, n\2, a[idx++]=n!/(2^k*(n-2*k)!)));
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