A353225 Expansion of e.g.f. (1 - x^4)^(-1/x^3).

1, 1, 1, 1, 1, 61, 361, 1261, 3361, 128521, 1678321, 11670121, 56596321, 1773048421, 37020623641, 410615985781, 3056256665281, 88439609228881, 2516514283997281, 39513591769228561, 409546654143301441, 11679302565962651341, 413008783534735181641

Offset

0,6

Links

Table of n, a(n) for n=0..22.

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=1..floor((n+3)/4)} (4*k-3)/k * a(n-4*k+3)/(n-4*k+3)!.

a(n) = n! * Sum_{k=0..floor(n/4)} |Stirling1(n-3*k,n-4*k)|/(n-3*k)!.

a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (4*exp(n)). - Vaclav Kotesovec, May 04 2022

Mathematica

With[{nn=30}, CoefficientList[Series[(1-x^4)^(-1/x^3), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Sep 17 2024 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^4)^(-1/x^3)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x^4)/x^3)))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, (i+3)\4, (4*j-3)/j*v[i-4*j+4]/(i-4*j+3)!)); v;
(PARI) a(n) = n!*sum(k=0, n\4, abs(stirling(n-3*k, n-4*k, 1))/(n-3*k)!);

Crossrefs

Cf. A121452, A353223, A353224.

Sequence in context: A361673 A373525 A357965 * A373520 A189174 A304062

Adjacent sequences: A353222 A353223 A353224 * A353226 A353227 A353228

Keyword

nonn

Author

Seiichi Manyama, May 01 2022

Status

approved