A335345 Expansion of e.g.f. exp(x^2/(2*(1 - x)^3)).

1, 0, 1, 9, 75, 690, 7305, 89145, 1237425, 19221300, 329371245, 6157738125, 124551652995, 2707913238030, 62945320162725, 1557291398788125, 40844991621859425, 1131753403094113800, 33025920511859300025, 1012128709342410284625, 32494107983067177522075

Offset

0,4

Links

Table of n, a(n) for n=0..20.

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A001809(k) * a(n-k).

D-finite with recurrence 2*a(n) +8*(-n+1)*a(n-1) +2*(n-1)*(6*n-13)*a(n-2) -(n-1)*(n-2)*(8*n-23)*a(n-3) +2*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 05 2020

a(n) ~ 2^(-9/8) * 3^(1/8) * n^(n - 1/8) * exp(1/54 - n^(1/4)/(2^(15/4)*3^(5/4)) - sqrt(6*n)/12 + 2^(7/4)*3^(-3/4)*n^(3/4) - n). - Vaclav Kotesovec, Jun 11 2020

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n+k-1,n-2*k)/(2^k * k!). - Seiichi Manyama, Jun 17 2024

Mathematica

nmax = 20; CoefficientList[Series[Exp[x^2/(2 (1 - x)^3)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (1/4) Sum[Binomial[n - 1, k - 1] k (k - 1) k! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Prog

(PARI) seq(n)=Vec(serlaplace(exp(x^2/(2*(1 - x)^3) + O(x*x^n)))) \\ Andrew Howroyd, Jun 02 2020

Crossrefs

Cf. A001809, A091695, A185369, A335344.

Sequence in context: A351513 A136659 A231592 * A126965 A066222 A080254

Adjacent sequences: A335342 A335343 A335344 * A335346 A335347 A335348

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 02 2020

Status

approved