A353225 - OEIS

%I #22 Sep 17 2024 13:22:32

%S 1,1,1,1,1,61,361,1261,3361,128521,1678321,11670121,56596321,

%T 1773048421,37020623641,410615985781,3056256665281,88439609228881,

%U 2516514283997281,39513591769228561,409546654143301441,11679302565962651341,413008783534735181641

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

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

%F a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (4*exp(n)). - _Vaclav Kotesovec_, May 04 2022

%t 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 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^4)^(-1/x^3)))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x^4)/x^3)))

%o (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 (PARI) a(n) = n!*sum(k=0, n\4, abs(stirling(n-3*k, n-4*k, 1))/(n-3*k)!);

%Y Cf. A121452, A353223, A353224.

%K nonn

%O 0,6

%A _Seiichi Manyama_, May 01 2022