A361573 Expansion of e.g.f. exp(x^3/(6 * (1 - x)^3)).

1, 0, 0, 1, 12, 120, 1210, 13020, 152880, 1975960, 28148400, 440470800, 7525441000, 139375236000, 2778421245600, 59239029249400, 1343609515248000, 32274288638592000, 818014942318974400, 21809788600885084800, 610079100418595808000, 17863467401461938256000

Offset

0,5

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-1,n-3*k)/(6^k * k!).

a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} (-1)^(k-3) * k * binomial(-3,k-3) * a(n-k)/(n-k)!.

a(n) ~ 2^(-9/8) * exp(-1/24 + 5*2^(1/4)*n^(1/4)/48 - sqrt(2*n)/4 + 2^(7/4)*n^(3/4)/3 - n) * n^(n - 1/8) * (1 - 1203*2^(3/4)/(10240*n^(1/4))). - Vaclav Kotesovec, Mar 29 2023

Mathematica

With[{nn=30}, CoefficientList[Series[Exp[x^3/(6(1-x)^3)], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, May 20 2023 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/(6*(1-x)^3))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=3, i, (-1)^(j-3)*j*binomial(-3, j-3)*v[i-j+1]/(i-j)!)); v;

Crossrefs

Cf. A000262, A335344, A361577.

Sequence in context: A037694 A242810 A067102 * A129064 A120585 A239188

Adjacent sequences: A361570 A361571 A361572 * A361574 A361575 A361576

Keyword

nonn

Author

Seiichi Manyama, Mar 16 2023

Status

approved