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

%I A353225 #22 Sep 17 2024 13:22:32
%S A353225 1,1,1,1,1,61,361,1261,3361,128521,1678321,11670121,56596321,
%T A353225 1773048421,37020623641,410615985781,3056256665281,88439609228881,
%U A353225 2516514283997281,39513591769228561,409546654143301441,11679302565962651341,413008783534735181641
%N A353225 Expansion of e.g.f. (1 - x^4)^(-1/x^3).
%F A353225 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)!.
%F A353225 a(n) = n! * Sum_{k=0..floor(n/4)} |Stirling1(n-3*k,n-4*k)|/(n-3*k)!.
%F A353225 a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (4*exp(n)). - _Vaclav Kotesovec_, May 04 2022
%t A353225 With[{nn=30},CoefficientList[Series[(1-x^4)^(-1/x^3),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Sep 17 2024 *)
%o A353225 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^4)^(-1/x^3)))
%o A353225 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x^4)/x^3)))
%o A353225 (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;
%o A353225 (PARI) a(n) = n!*sum(k=0, n\4, abs(stirling(n-3*k, n-4*k, 1))/(n-3*k)!);
%Y A353225 Cf. A121452, A353223, A353224.
%K A353225 nonn
%O A353225 0,6
%A A353225 _Seiichi Manyama_, May 01 2022