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