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Search: a000166 -id:a000166
Displaying 1-10 of 538 results found.
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A082491
a(n) = n! * d(n), where n! = factorial numbers (A000142), d(n) = subfactorial numbers (A000166).
+20
10
1, 0, 2, 12, 216, 5280, 190800, 9344160, 598066560, 48443028480, 4844306476800, 586161043776000, 84407190782745600, 14264815236056985600, 2795903786354347468800, 629078351928420506112000, 161044058093696572354560000, 46541732789077953723039744000
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
a(n) is also the number of pairs of n-permutations p and q such that p(x)<>q(x) for each x in { 1, 2, ..., n }.
Or number of n X n matrices with exactly one 1 and one 2 in each row and column, other entries 0 (cf. A001499). - Vladimir Shevelev, Mar 22 2010
a(n) is approximately equal to (n!)^2/e. - J. M. Bergot, Jun 09 2018
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Ira Gessel, Enumerative applications of symmetric functions, Séminaire Lotharingien de Combinatoire, B17a (1987), 17 pp.
Shawn L. Witte, Link Nomenclature, Random Grid Diagrams, and Markov Chain Methods in Knot Theory, Ph. D. Dissertation, University of California-Davis (2020).
FORMULA
a(n) = n! * d(n) where d(n) = A000166(n).
a(n) = Sum_{k=0..n} binomial(n, k)^2 * (-1)^k * (n - k)!^2 * k!.
a(n+2) = (n+2)*(n+1) * ( a(n+1) + (n+1)*a(n) ).
a(n) ~ 2*Pi*n^(2*n+1)*exp(-2*n-1). - Ilya Gutkovskiy, Dec 04 2016
MAPLE
with (combstruct):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(2):seq(count(ZLL, size=n)*n!, n=0..15); # Zerinvary Lajos, Jun 11 2008
MATHEMATICA
Table[Subfactorial[n]*n!, {n, 0, 15}] (* Zerinvary Lajos, Jul 10 2009 *)
PROG
(Maxima) A000166[0]:1$
A000166[n]:=n*A000166[n-1]+(-1)^n$
makelist(n!*A000166[n], n, 0, 12); /* Emanuele Munarini, Mar 01 2011 */
(PARI)
d(n)=if(n<1, n==0, n*d(n-1)+(-1)^n);
a(n)=d(n)*n!;
vector(33, n, a(n-1))
/* Joerg Arndt, May 28 2012 */
(PARI) {a(n) = if( n<2, n==0, n! * round(n! / exp(1)))}; /* Michael Somos, Jun 24 2018 */
(Python)
A082491_list, m, x = [], 1, 1
for n in range(10*2):
....x, m = x*n**2 + m, -(n+1)*m
....A082491_list.append(x) # Chai Wah Wu, Nov 03 2014
(Scala)
val A082491_pairs: LazyList[BigInt && BigInt] =
(BigInt(0), BigInt(1)) #::
(BigInt(1), BigInt(0)) #::
lift2 {
case ((n, z), (_, y)) =>
(n+2, (n+2)*(n+1)*((n+1)*z+y))
} (A082491_pairs, A082491_pairs.tail)
val A082491: LazyList[BigInt] =
lift1(_._2)(A082491_pairs)
/** Luc Duponcheel, Jan 25 2020 */
CROSSREFS
Cf. A000142, A000166.
KEYWORD
easy,nonn
AUTHOR
Emanuele Munarini, Apr 28 2003
STATUS
approved
A101559
This table (read by rows) shows the coefficients of sum formulas of n-th subfactorial numbers (A000166). The n-th row (n>=1) contains T(i,n) for i=1 to n, where T(i,n) satisfies Subf(n) = Sum_{i=1..n} T(i,n) * n^(n-i).
+20
7
1, 1, -2, 1, -4, 4, 1, -7, 15, -10, 1, -11, 42, -65, 34, 1, -16, 96, -267, 339, -154, 1, -22, 191, -831, 1891, -2103, 874, 1, -29, 344, -2151, 7600, -15023, 15171, -5914, 1, -37, 575, -4880, 24600, -74884, 133147, -124755, 46234, 1, -46, 907, -10025, 68153, -293925, 798564, -1305847, 1151331, -409114, 1, -56
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Table of n, a(n) for n=1..57.
EXAMPLE
Subf(9) = [ 9^8 -37*9^7 +575*9^6 -4880*9^5 +24600*9^4 -74884*9^3 +133147*9^2 - 124755*9 +46234 ] = 14833.
CROSSREFS
Cf. A101560, A000166, A000110, A101033, A101032, A000204, A100492, A099731, A000045, A094216, A094638, A000108.
KEYWORD
sign,tabl
AUTHOR
André F. Labossière, Dec 06 2004
STATUS
approved
A334715
A(n,k) = !n + [n > 0] * (k * n!), where !n = A000166(n) is subfactorial of n and [] is an Iverson bracket; square array A(n,k), n>=0, k>=0, read by antidiagonals.
+20
6
1, 1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 3, 5, 8, 9, 1, 4, 7, 14, 33, 44, 1, 5, 9, 20, 57, 164, 265, 1, 6, 11, 26, 81, 284, 985, 1854, 1, 7, 13, 32, 105, 404, 1705, 6894, 14833, 1, 8, 15, 38, 129, 524, 2425, 11934, 55153, 133496, 1, 9, 17, 44, 153, 644, 3145, 16974, 95473, 496376, 1334961
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Derangement
Wikipedia, Iverson bracket
FORMULA
E.g.f. of column k: (k*exp(x)*x+1)*exp(-x)/(1-x).
A(n,k) = A000166(n) + [n > 0] * (k * n!).
A(n,k) = (k-1)*n + 1 if n<2, A(n,k) = n*A(n-1, k) + (-1)^n if n>=2.
EXAMPLE
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, ...
1, 3, 5, 7, 9, 11, 13, 15, ...
2, 8, 14, 20, 26, 32, 38, 44, ...
9, 33, 57, 81, 105, 129, 153, 177, ...
44, 164, 284, 404, 524, 644, 764, 884, ...
265, 985, 1705, 2425, 3145, 3865, 4585, 5305, ...
1854, 6894, 11934, 16974, 22014, 27054, 32094, 37134, ...
...
MAPLE
A:= proc(n, k) option remember; `if`(n<2,
(k-1)*n+1, n*A(n-1, k)+(-1)^n)
end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
MATHEMATICA
A[n_, k_] := Subfactorial[n] + Boole[n>0] k n!;
Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)
CROSSREFS
Columns k=0-3 give: A000166, A001120, A110043, A110149.
Rows n=0-3 give: A000012, A001477, A005408, A016933.
Main diagonal gives A334716.
Cf. A000142.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 08 2020
STATUS
approved
A378157
The least prime dividing !n = A000166(n).
+20
5
2, 3, 2, 5, 2, 7, 2, 3, 2, 11, 2, 13, 2, 3, 2, 17, 2, 11, 2, 3, 2, 23, 2, 5, 2, 3, 2, 29, 2, 31, 2, 3, 2, 5, 2, 11, 2, 3, 2, 11, 2, 43, 2, 3, 2, 47, 2, 7, 2, 3, 2, 53, 2, 5, 2, 3, 2, 11, 2, 61, 2, 3, 2, 5, 2, 67, 2, 3, 2, 71, 2, 73, 2, 3, 2, 7, 2, 79, 2, 3, 2
(list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
LINKS
Amiram Eldar, Table of n, a(n) for n = 3..10000
FORMULA
a(n) = A020639(A000166(n)).
a(n) = min(A020639(n-1), A378159(n-2)) for n >= 2.
MATHEMATICA
lpf[n_] := Module[{p = 2}, While[! Divisible[n, p], p = NextPrime[p]]; Array[lpf[Subfactorial[#]] &, 50, 3]
PROG
(PARI) lpf(n) = {my(p = 2); while(n % p, p = nextprime(p+1)); p; }
lista(nmax) = {my(s = 1); for(n = 3, nmax, s = n * s + (-1)^n; print1(lpf(s), ", ")); }
CROSSREFS
Cf. A000166, A020639, A152024, A195207, A195208, A195209, A195210, A378158, A378159.
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 18 2024
STATUS
approved
A065087
a(n) = A000166(n)*binomial(n+1,2).
+20
4
0, 0, 3, 12, 90, 660, 5565, 51912, 533988, 6007320, 73422855, 969181620, 13744757598, 208462156812, 3367465610145, 57727981888080, 1046800738237320, 20020064118788592, 402756584036805963, 8502638996332570140, 187953072550509445410, 4341715975916768188740
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
a(n) is also the number of permutations of [2n-1] having n-1 isolated fixed points (i.e. adjacent entries are not fixed points). Example: a(2)=3 because we have 132, 213, and 321. - Emeric Deutsch, Apr 18 2009
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..400
FORMULA
a(n) = (n/2)*A000240(n+1). - Zerinvary Lajos, Dec 18 2007, corrected Jul 09 2012
a(n) = n * (n+1) * (a(n-1)/(n-1) + (-1)^n/2) for n > 1 - Seiichi Manyama, Jun 24 2018
E.g.f.: exp(-x)*x^2*(3 - 2*x + x^2)/(2*(1 - x)^3). - Ilya Gutkovskiy, Jun 25 2018
MATHEMATICA
a[n_] := Subfactorial[n]*Binomial[n + 1, 2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 18 2024 *)
CROSSREFS
Equals 3 * A000313(n+2).
Cf. A000166, A000240, A000387, A305730.
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 10 2001
STATUS
approved
A173184
Partial sums of A000166.
+20
4
1, 1, 2, 4, 13, 57, 322, 2176, 17009, 150505, 1485466, 16170036, 192384877, 2483177809, 34554278858, 515620794592, 8212685046337, 139062777326001, 2494364438359954, 47245095998005060, 942259727190907181, 19737566982241851721, 433234326593362631602
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Partial sums of subfactorial or rencontres numbers, or derangements (number of permutations of n elements with no fixed points). The subsequence of primes begins: 2, 13, 192384877.
LINKS
Table of n, a(n) for n=0..22.
FORMULA
G.f.: 1/U(0)/(1-x) where U(k) = 1 + x - x*(k+1)/(1 - x*(k+1)/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 15 2012
G.f.: 1/(1 - x^2) + (1/(1 - x))*Sum_{k>=1} k^k*x^k/(1 + (k + 1)*x)^(k+1). - Ilya Gutkovskiy, Apr 13 2017
From Emanuele Munarini, Oct 06 2017: (Start)
E.g.f.: exp(-t)/(1-t) - exp(t-2)*(coshIntegral(2-2*t) + sinhIntegral(2-2*t) - expIntegralEi(2)).
a(n+2) - (n+3)*a(n+1) + (n+2)*a(n) = (-1)^n. (End)
D-finite with recurrence a(n+3) - (n+3)*a(n+2) + (n+2)*a(n) = 0. - Emanuele Munarini, Aug 24 2017
EXAMPLE
a(3) = 1 + 0 + 1 + 2 = 4.
MATHEMATICA
a[0] = 1; a[n_] := a[n] = n*a[n - 1] + (-1)^n; Accumulate@ Array[a, 21, 0] (* Robert G. Wilson v, Apr 01 2011 *)
dr[{n_, a1_, a2_}]:={n+1, a2, n(a1+a2)}; Accumulate[Transpose[NestList[dr, {0, 0, 1}, 30]][[3]]] (* Harvey P. Dale, Jul 17 2014 *)
Table[Sum[Subfactorial[k], {k, 0, n}], {n, 0, 100}] (* Emanuele Munarini, Aug 24 2017 *)
PROG
(Maxima)
s[0]:1$
s[n]:=n*s[n-1]+(-1)^n$
makelist(sum(s[k], k, 0, n), n, 0, 12); /* Emanuele Munarini, Aug 24 2017 */
CROSSREFS
Cf. A000166, A001277.
KEYWORD
nonn
AUTHOR
Jonathan Vos Post, Feb 12 2010
STATUS
approved
A159610
Triangle read by rows, n-th row = n terms of A000255: (1, 3, 11, 53, 309, ...); right border = A000166 starting (1, 2, 9, 44, 265, ...).
+20
3
1, 3, 2, 11, 11, 9, 53, 53, 53, 44, 309, 309, 309, 309, 265, 2119, 2119, 2119, 2119, 2119, 1854, 16687, 16687, 16687, 16687, 16687, 14833, 148329, 148329, 148329, 148329, 148329, 148329, 148329, 133496
(list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Row sums = A002469(n+2), representing the game of mousetrap with n cards; where nonzero terms of A002469 start: (1, 5, 31, 203, 1501, ...). A002469(n) = (n-2)*A000255(n-1) + A000166(n). Example 31 = 2*11 + 9 = A002469(4) = 2*A000255(3) + A000166(4).
LINKS
Table of n, a(n) for n=0..34.
FORMULA
Triangle read by rows, n-th row = n terms of A000255: (1, 3, 11, 53, 309, ...); right border = A000166 starting (1, 2, 9, 44, 265, ...)
EXAMPLE
First few rows of the triangle:
1;
3, 2;
11, 11, 9;
53, 53, 53, 44;
309, 309, 309, 309, 265;
2119, 2119, 2119, 2119, 2119, 1854;
16687, 16687, 16687, 16687, 16687, 16687, 14833;
...
CROSSREFS
Cf. A002469, A000166, A000255.
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Apr 17 2009
STATUS
approved
A301423
Numbers k such that !k/(k-1) is prime, where !k = A000166(k) is the subfactorial of k.
+20
3
4, 5, 6, 11, 15, 44, 66, 168, 575, 1713
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Also numbers k such that A000255(k-2) is prime.
The corresponding primes are 3, 11, 53, 1468457, 34361893981, 22742406079421034331584846001936724930824184898296683, ...
a(11) > 35000. - Robert Price, Apr 14 2018
LINKS
Table of n, a(n) for n=1..10.
MATHEMATICA
Select[Range[2, 100], PrimeQ[Subfactorial[#]/(#-1)] &]
PROG
(PARI) isok(n) = (n != 1) && isprime(n!*sum(k=0, n, (-1)^k/k!)/(n-1)); \\ Michel Marcus, Mar 24 2018
CROSSREFS
Cf. A000166, A000255.
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Mar 20 2018
STATUS
approved
A378160
The number of distinct prime factors of !n = A000166(n).
+20
3
0, 1, 1, 2, 2, 3, 3, 4, 2, 3, 2, 4, 3, 3, 4, 3, 3, 4, 5, 5, 4, 7, 5, 6, 4, 5, 7, 6, 6, 7, 4, 4, 4, 8, 4, 6, 4, 5, 6, 6, 4, 7, 2, 4, 7, 8, 6, 5, 7, 6, 7, 7, 4, 6, 9, 6, 6, 6, 6, 6, 4, 4, 5, 4, 3, 6, 6, 6, 6, 6, 7, 7, 4, 8, 6, 5, 8, 6, 4, 4, 5, 8, 4, 7, 7, 8, 6
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,4
LINKS
Sean A. Irvine, Table of n, a(n) for n = 2..108
FORMULA
a(n) = A001221(A000166(n)).
a(n) >= A001221(n-1) + 1 for n >=5.
MATHEMATICA
Array[PrimeNu[Subfactorial[#]] &, 40, 2]
PROG
(PARI) lista(nmax) = {my(s = 0); for(n = 2, nmax, s = n * s + (-1)^n; print1(omega(s), ", ")); }
CROSSREFS
Cf. A000166, A001221, A152024, A195207, A195208, A195209, A195210, A301423, A378157, A378161, A378162.
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 18 2024
EXTENSIONS
a(82)-a(88) from Jinyuan Wang, Nov 24 2024
STATUS
approved
A378161
The number of prime factors of !n = A000166(n), counted with multiplicity.
+20
3
0, 1, 2, 3, 2, 4, 3, 6, 5, 3, 3, 6, 3, 3, 5, 6, 3, 8, 5, 6, 5, 8, 5, 9, 5, 5, 11, 7, 6, 9, 4, 8, 6, 8, 4, 10, 5, 5, 7, 8, 4, 8, 2, 7, 12, 8, 6, 9, 8, 7, 8, 8, 4, 10, 10, 8, 7, 6, 6, 8, 4, 4, 8, 9, 3, 8, 6, 7, 7, 6, 7, 13, 4, 8, 8, 6, 9, 7, 4, 7, 10, 8, 4, 9, 7
(list; graph; refs; listen; history; text; internal format)
OFFSET
2,3
LINKS
Sean A. Irvine, Table of n, a(n) for n = 2..108
FORMULA
a(n) = A001222(A000166(n)).
a(n) >= A001222(n-1) + 1 for n >=4, with equality if and only if n is in A301423.
MATHEMATICA
Array[PrimeOmega[Subfactorial[#]] &, 40, 2]
PROG
(PARI) lista(nmax) = {my(s = 0); for(n = 2, nmax, s = n * s + (-1)^n; print1(bigomega(s), ", ")); }
CROSSREFS
Cf. A001222, A000166, A152024, A195207, A195208, A195209, A195210, A301423, A378157, A378160, A378162.
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 18 2024
EXTENSIONS
a(82)-a(86) from Jinyuan Wang, Nov 24 2024
STATUS
approved
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