A000090 -id:A000090 - cp's OEIS frontend

A000266 Expansion of e.g.f. exp(-x^2/2) / (1-x).

1, 1, 1, 3, 15, 75, 435, 3045, 24465, 220185, 2200905, 24209955, 290529855, 3776888115, 52876298475, 793144477125, 12690313661025, 215735332237425, 3883235945814225, 73781482970470275, 1475629660064134575, 30988222861346826075, 681740902935880863075

Offset: 0

N. J. A. Sloane

nonn

a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no transposition.

			a(3) = 3 because the permutations in S_3 that contain no transpositions are the trivial permutation and the two 3-cycles.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 85.
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).
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
Larry Carter and Stan Wagon, The Mensa Correctional Institute, The American Mathematical Monthly 125.4 (2018): 306-319.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 104
Simon Plouffe, Exact formulas for integer sequences, March 1993.

A000138 Expansion of e.g.f. exp(-x^4/4)/(1-x).

Original entry on oeis.org

1, 1, 2, 6, 18, 90, 540, 3780, 31500, 283500, 2835000, 31185000, 372972600, 4848643800, 67881013200, 1018215198000, 16294848570000, 277012425690000, 4986223662420000, 94738249585980000, 1894745192712372000, 39789649046959812000, 875372279033115864000

Offset: 0

N. J. A. Sloane

a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no 4-cycle.

			a(4) = 18 because in S_4 the permutations with no 4-cycle are the complement of the six 4-cycles so a(4) = 4! - 6 = 18.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 85.
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).
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.

T. D. Noe, Table of n, a(n) for n = 0..100
Simon Plouffe, Exact formulas for integer sequences

Cf. A000142, A000090, A000266, A060725, A060726, A060727.

nn=20;Range[0,nn]!CoefficientList[Series[Exp[-x^4/4]/(1-x),{x,0,nn}],x]  (* Geoffrey Critzer, Oct 28 2012 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( -(x^4/4) + x*O(x^n)) / (1 - x), n))} /* Michael Somos, Jul 28 2009 */

a(n) = n! * Sum_{i=0..floor(n/4)} (-1)^i / (i! * 4^i); a(n)/n! ~ Sum_{i >= 0} (-1)^i / (i! * 4^i) = e^(-1/4); a(n) ~ e^(-1/4) * n!; a(n) ~ e^(-1/4) * (n/e)^n * sqrt(2*Pi*n). - Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001
a(n,k) = n!*floor(floor(n/k)!*k^floor(n/k)/exp(1/k) + 1/2)/(floor(n/k)!*k^floor(n/k)), here k=4, n>=0. Simon Plouffe, from old notes, 1993
E.g.f.: exp(-x^4/4)/(1-x) = 1/G(0); G(k) = 1 - x/(1 - (x^3)/(x^3 - 4*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Feb 28 2012

Entry improved by comments from Michael Somos, Jul 28 2009

Name corrected by Joerg Arndt, May 27 2011

A060725 E.g.f.: exp(-(x^5/5))/(1-x).

Original entry on oeis.org

1, 1, 2, 6, 24, 96, 576, 4032, 32256, 290304, 2975616, 32731776, 392781312, 5106157056, 71486198784, 1070549415936, 17128790654976, 291189441134592, 5241409940422656, 99586788868030464, 1991897970827821056, 41829857387384242176, 920256862522453327872

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001

nonn

a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no 5-cycle.

			a(5) = 96 because in S_5 the permutations with no 5-cycle are the complement of the 24 5-cycles so a(5) = 5! - 24 = 96.

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.

Harry J. Smith, Table of n, a(n) for n = 0..100
Plouffe, Simon, Exact formulas for integer sequences

Cf. A000142, A000090, A000138, A000266, A060725, A060726, A060727.

for n from 0 to 30 do printf(`%d,`, n! * sum(( (-1)^i /(i! * 5^i)), i=0..floor(n/5))) od:

With[{nn=30},CoefficientList[Series[Exp[-(x^5/5)]/(1-x),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 24 2019 *)

{ for (n=0, 100, write("b060725.txt", n, " ", n! * sum(i=0, n\5, (-1)^i / (i! * 5^i))); ) } \\ Harry J. Smith, Jul 10 2009

{a(n) = if( n<0, 0, n! * polcoeff( exp( -(x^5 / 5) + x*O(x^n)) / (1 - x), n))} /* Michael Somos, Jul 28 2009 */

{ A060725_list(numterms) = Vec(serlaplace(exp(-x^5/5 + O(x^numterms))/(1-x))); } /* Eric M. Schmidt, Aug 22 2012 */

The formula for a(n) is: a(n) = n! * sum i=0 ... [ n/5 ]( (-1)^i /(i! * 5^i)) by this formula we have as n -> infinity: a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 5^i) = e^(-1/5) or a(n) ~ e^(-1/5) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/5) * (n/e)^n * sqrt(2 * Pi * n).

a(n,k) = n!*floor(floor(n/k)!*k^floor(n/k)/exp(1/k) + 1/2)/(floor(n/k)! * k^floor(n/k)), k=5, n>=0. - Simon Plouffe, Feb 18 2011

More terms from James Sellers, Apr 24 2001

Entry improved by comments from Michael Somos, Jul 28 2009

A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 1, 10, 478, 68248, 21809656, 13107532816, 13244650672240, 20818058883902848, 48069880140604832128, 156044927762422185270016, 687740710497308621254625536, 4000181720339888446834235653120, 29991260979682976913756629498334208

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The multinomial transform [MNL] transforms an input sequence b(n) into the output sequence a(n). Given the fact that the structure of the a(n) formulas, see the examples, lead to the multinomial coefficients A036039 the MNL transform seems to be an appropriate name for this transform. The multinomial transform is related to the exponential transform, see A274804 and the third formula. For the inverse multinomial transform [IML] see A274844.
To preserve the identity IML[MNL[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A036039 appear.
We observe that a(0) = 1 and that this term provides no information about any value of b(n), this notwithstanding we will start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the multinomial transform of a sequence. The first program uses the first formula which was found by Paul D. Hanna, see A158876, and Vladimir Kruchinin, see A215915. The second program uses properties of the e.g.f., see the sequences A158876, A213507, A244430 and A274539 and the third formula. The third program uses information about the inverse multinomial transform, see A274844.
Some MNL transform pairs are, n >= 1: A000045(n) and A244430(n-1); A000045(n+1) and A213527(n-1); A000108(n) and A213507(n-1); A000108(n-1) and A243953(n-1); A000142(n) and A158876(n-1); A000203(n) and A053529(n-1); A000110(n) and A274539(n-1); A000041(n) and A215915(n-1); A000035(n-1) and A177145(n-1); A179184(n) and A038205(n-1); A267936(n) and A000266(n-1); A267871(n) and A000090(n-1); A193356(n) and A088009(n-1).

			Some a(n) formulas, see A036039:
  a(0) = 1
  a(1) = 1*x(1)
  a(2) = 1*x(2) + 1*x(1)^2
  a(3) = 2*x(3) + 3*x(1)*x(2) + 1*x(1)^3
  a(4) = 6*x(4) + 8*x(1)*x(3) + 3*x(2)^2 + 6*x(1)^2*x(2) + 1*x(1)^4
  a(5) = 24*x(5) + 30*x(1)*x(4) + 20*x(2)*x(3) + 20*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 10*x(1)^3*x(2) + 1*x(1)^5

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, arXiv:math/0205301 [math.CO], 2002; Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.
Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.
Cf. A090998, A008298, A271703, A112302, A135002, A274540.

nmax:= 13: b := proc(n): (doublefactorial(2*n-1))^2 end: a:= proc(n) option remember: if n=0 then 1 else add(((n-1)!/(n-k)!) * b(k) * a(n-k), k=1..n) fi: end: seq(a(n), n = 0..nmax); # End first MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := exp(add(b(n)*x^n/n, n = 1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n = 0..nmax); # End second MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(log(1+add(s(n)*x^n/n!, n=1..nmax)), x, nmax+1): d := proc(n): n*coeff(f, x, n) end: a(0) := 1: a(1) := b(1): s(1) := b(1): for n from 2 to nmax do s(n) := solve(d(n)-b(n), s(n)): a(n):=s(n): od: seq(a(n), n=0..nmax); # End third MNL program.

b[n_] := (2*n - 1)!!^2;
a[0] = 1; a[n_] := a[n] = Sum[((n-1)!/(n-k)!)*b[k]*a[n-k], {k, 1, n}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 17 2017 *)

a(n) = Sum_{k=1..n} ((n-1)!/(n-k)!)*b(k)*a(n-k), n >= 1 and a(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
a(n) = n!*P(n), with P(n) = (1/n)*(Sum_{k=0..n-1} b(n-k)*P(k)), n >= 1 and P(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n) with b(n) = A001818(n) = ((2*n-1)!!)^2.
denom(a(n)/2^n) = A001316(n); numer(a(n)/2^n) = [1, 1, 5, 239, 8531, 2726207, ...].

A060726 For n >= 1, a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 6-cycle.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 600, 4200, 33600, 302400, 3024000, 33264000, 405820800, 5275670400, 73859385600, 1107890784000, 17726252544000, 301346293248000, 5419293175296000, 102966570330624000, 2059331406612480000

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001

nonn

This is the expansion of exp ((-x^6)/6) /(1-x).

			a(6) = 600 because in S_6 the permutations with no 6-cycle are the complement of the 120 6-cycles so a(6) = 6! - 120 = 600.

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.

Harry J. Smith, Table of n, a(n) for n = 0..100
Plouffe, Simon, Exact formulas for integer sequences

Cf. A000142, A000266, A000090, A000138, A060725, A060726, A060727.

for n from 0 to 30 do printf(`%d,`, n! * sum(( (-1)^i /(i! * 6^i)), i=0..floor(n/6))) od:

a(n)={n! * sum(i=0, n\6, (-1)^i / (i! * 6^i))} \\ Harry J. Smith, Jul 10 2009

The formula for a(n) is: a(n) = n! * Sum_{i=0..floor(n/6)} ((-1)^i /(i! * 6^i)) by this formula we have as n -> infinity: a(n)/n! ~ Sum_{i>= 0} (-1)^i /(i! * 6^i) = e^(-1/6) or a(n) ~ e^(-1/6) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/6) * (n/e)^n * sqrt(2 * Pi * n)

a(n,k) = n!*floor(floor(n/k)!*k^floor(n/k)/exp(1/k) + 1/2)/(floor(n/k)!*k^floor(n/k)), k=6, n >= 0. - Simon Plouffe, Feb 18 2011

More terms from James Sellers, Apr 24 2001

A114320 Triangle T(n,k) = number of permutations of n elements with k 2-cycles.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 15, 6, 3, 75, 30, 15, 435, 225, 45, 15, 3045, 1575, 315, 105, 24465, 12180, 3150, 420, 105, 220185, 109620, 28350, 3780, 945, 2200905, 1100925, 274050, 47250, 4725, 945, 24209955, 12110175, 3014550, 519750, 51975, 10395, 290529855

Offset: 0

Vladeta Jovovic, Feb 05 2006

Row n has 1+floor(n/2) terms. Row sums yield the factorials (A000142). Sum(k*T(n,k),k>0)=n!/2 for n>=2. - Emeric Deutsch, Feb 17 2006

			T(3,1) = 3 because we have (1)(23), (12)(3) and (13)(2).
Triangle begins:
    1;
    1;
    1,   1;
    3,   3;
   15,   6,   3;
   75,  30,  15;
  435, 225,  45,  15;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A008290, A000266, A000090, A088436, A000138, A060725, A060726, A086659, A105422, A105114.

G:= exp((y-1)*x^2/2)/(1-x): Gser:= simplify(series(G,x=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:= n!*coeff(Gser,x^n) od: for n from 0 to 12 do seq(coeff(y*P[n], y^j), j=1..1+floor(n/2)) od;  # yields sequence in triangular form - Emeric Deutsch, Feb 17 2006

d = Exp[-x^2/2!]/(1 - x);f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Transpose[Table[Range[0, 10]!CoefficientList[Series[x^(2 k)/(2^k k!) d, {x, 0, 10}], x], {k, 0, 5}]]]]  (* Geoffrey Critzer, Nov 29 2011 *)

E.g.f.: exp((y-1)*x^2/2)/(1-x). More generally, e.g.f. for number of permutations of n elements with k m-cycles is exp((y-1)*x^m/m)/(1-x).

T(n,k) = n!/(2^k*k!) * Sum_{j=0..floor(n/2)-k} (-1/2)^j/j!. - Alois P. Heinz, Nov 30 2011

More terms from Emeric Deutsch, Feb 17 2006

A060727 For n >= 1 a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 7-cycle.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 4320, 34560, 311040, 3110400, 34214400, 410572800, 5337446400, 75613824000, 1134207360000, 18147317760000, 308504401920000, 5553079234560000, 105508505456640000, 2110170109132800000, 44288746761093120000, 974352428744048640000

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001

nonn

This is the expansion of exp ((-x^7)/7)/(1-x).

			a(7) = 4320 because in S_7 the permutations with no 7-cycle are the complement of the 720 7-cycles so a(7) = 7! - 720 = 4320.

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A000142, A000090, A000266, A000138, A060725, A060726, A060727.

for n from 0 to 30 do printf(`%d,`, n! * sum(( (-1)^i /(i! * 7^i)), i=0..floor(n/7))) od:

{ for (n=0, 100, write("b060727.txt", n, " ", n! * sum(i=0, n\7, (-1)^i / (i! * 7^i))); ) } \\ Harry J. Smith, Jul 10 2009

The formula for a(n) is: a(n) = n! * sum i=0 ... [ n/7 ]( (-1)^i /(i! * 7^i)) by this formula we have as n -> infinity: a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 7^i) = e^(-1/7) or a(n) ~ e^(-1/7) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/7) * (n/e)^n * sqrt(2 * Pi * n)

a(n,k) = n!*floor(floor(n/k)!*k^floor(n/k)/exp(1/k) + 1/2)/(floor(n/k)!*k^floor(n/k)), k=7, n>=0. - Simon Plouffe, Feb 18 2011

More terms from James Sellers, Apr 24 2001

A102736 Number of permutations of n elements without cycles whose length is a multiple of 3.

Original entry on oeis.org

1, 1, 2, 4, 16, 80, 400, 2800, 22400, 179200, 1792000, 19712000, 216832000, 2818816000, 39463424000, 552487936000, 8839806976000, 150276718592000, 2554704216064000, 48539380105216000, 970787602104320000, 19415752042086400000, 427146544925900800000, 9824370533295718400000, 225960522265801523200000, 5649013056645038080000000, 146874339472770990080000000, 3818732826292045742080000000

Offset: 0

Vladeta Jovovic, Feb 08 2005

Differs from A247007 first at n=27. - Alois P. Heinz, Sep 09 2014

			G.f. = 1 + x + 2*x^2 + 4*x^3 + 16*x^4 + 80*x^5 + 400*x^6 + 2800*x^7 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000090, A000246.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
      irem(j, 3)=0, 0, a(n-j)*(j-1)!*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 31 2017

nn=21;a=Sum[x^n/n,{n,3,nn,3}];Range[0,nn]!CoefficientList[Series[Exp[Log[1/(1-x)]-a],{x,0,nn}],x]  (* Geoffrey Critzer, Nov 11 2012 *)
a[ n_] := If[ n < 0, 0, n! With[{m = Quotient[n, 3]}, (-1)^m Binomial[-2/3, m]]]; (* Michael Somos, Aug 05 2016 *)

{a(n) = my(m); if( n<0, 0, m = n\3; n! * (-1)^m * binomial(-2/3, m))}; /* Michael Somos, Aug 05 2016 */

E.g.f.: (1-x^3)^(1/3)/(1-x).

a(n) ~ n! * 3^(1/3) / (GAMMA(2/3) * n^(1/3)). - Vaclav Kotesovec, Mar 15 2014

A122974 Triangle T(n,k), the number of permutations on n elements that have no cycles of length k.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 9, 15, 16, 18, 44, 75, 80, 90, 96, 265, 435, 520, 540, 576, 600, 1854, 3045, 3640, 3780, 4032, 4200, 4320, 14833, 24465, 29120, 31500, 32256, 33600, 34560, 35280, 133496, 220185, 259840, 283500, 290304, 302400, 311040, 317520, 322560

Offset: 1

Dennis P. Walsh, Oct 27 2006

Read as sequence, a(n) is the number of permutations on j elements with no cycles of length i where j=round((2*n)^.5) and i=n-C(j,2).

T(n,k) generalizes several sequences already in the On-Line Encyclopedia, such as A000166, the number of permutations on n elements with no fixed points and A000266, the number of permutations on n elements with no transpositions (i.e., no 2-cycles). See the cross references for further examples.

			T(3,2)=3 since there are exactly 3 permutations of 1,2,3 that have no cycles of length 2, namely, (1)(2)(3),(1 2 3) and (2 1 3).
Triangle T(n,k) begins:
      0;
      1,     1;
      2,     3,     4;
      9,    15,    16,    18;
     44,    75,    80,    90,    96;
    265,   435,   520,   540,   576,   600;
   1854,  3045,  3640,  3780,  4032,  4200,  4320;
  14833, 24465, 29120, 31500, 32256, 33600, 34560, 35280;
  ...

Alois P. Heinz, Rows for n = 1..141, flattened
Dennis P. Walsh, The Number of Permutations with No k-Cycles.

Cf. T(n, 1)=A000166 for n=>1 T(n, 2)=A000266 for n=>2 T(n, 3)=A000090 for n=>3 T(n, 4)=A000138 for n=>4 T(n, 5)=A060725 for n=>5 T(n, 6)=A060726 for n=>6 T(n, 7)=A060727 for n=>7.

T(n,n) gives A094304(n+1).

seq((round((2*n)^.5))!*sum((-1/(n-binomial(round((2*n)^.5),2)))^r/r!,r=0..floor(round((2*n)^.5)/(n-binomial(round((2*n)^.5),2)))),n=1..66);
# second Maple program:
T:= proc(n, k) option remember; `if`(n=0, 1, add(`if`(j=k, 0,
      T(n-j, k)*binomial(n-1, j-1)*(j-1)!), j=1..n))
    end:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Nov 24 2019

T[n_, k_] := T[n, k] = If[n==0, 1, Sum[If[j==k, 0, T[n - j, k] Binomial[n - 1, j - 1] (j - 1)!], {j, 1, n}]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 08 2019, after Alois P. Heinz *)

T(n,k)=n!*sum r=0..floor(n/k)((-1/k)^r/r!) E.G.F: exp(-x^k/k)/(1-x) a(n)=(round((2*n)^.5))!*sum((-1/(n-binomial(round((2*n)^.5),2)))^r/r!,r=0..floor(round((2*n)^.5)/(n-binomial(round((2*n)^.5),2)))).

T(n,k) = n! - A293211(n,k). - Alois P. Heinz, Nov 24 2019

A319364 Expansion of e.g.f. exp(x^3/3)/(1 - x).

Original entry on oeis.org

1, 1, 2, 8, 32, 160, 1000, 7000, 56000, 506240, 5062400, 55686400, 668483200, 8690281600, 121663942400, 1825003980800, 29200063692800, 496401082777600, 8935231687782400, 169769402067865600, 3395388041357312000, 71303153503662080000, 1568669377080565760000, 36079395672853012480000

Offset: 0

Ilya Gutkovskiy, Sep 17 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..449

Cf. A000090, A000522, A030977, A052502, A130905, A319365.

seq(n!*coeff(series(exp(x^3/3)/(1 - x),x=0,24),x,n),n=0..23); # Paolo P. Lava, Jan 09 2019

nmax = 23; CoefficientList[Series[Exp[x^3/3]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

my(x='x+O('x^30)); Vec(serlaplace(exp(x^3/3)/(1 - x))) \\ Michel Marcus, Dec 17 2020

a(n) ~ n!*exp(1/3).

D-finite with recurrence: n*a(n) - n^2*a(n-1) - n*(n-1)*(n-2)*a(n-3) + n*(n-1)*(n-2)*(n-3)*a(n-4) = 0. - Robert Israel, Dec 17 2020

cp's OEIS Frontend

A000266 Expansion of e.g.f. exp(-x^2/2) / (1-x).

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A000138 Expansion of e.g.f. exp(-x^4/4)/(1-x).

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A060725 E.g.f.: exp(-(x^5/5))/(1-x).

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A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

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A060726 For n >= 1, a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 6-cycle.

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A114320 Triangle T(n,k) = number of permutations of n elements with k 2-cycles.

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