A382652 Expansion of e.g.f. exp( x/(1-3*x)^(5/3) ).
1, 1, 11, 151, 2601, 54401, 1341571, 38115351, 1225252561, 43935295681, 1737463744251, 75075845199191, 3517448555579641, 177538212306653121, 9600694935999031411, 553606933661659742551, 33899768045328467219361, 2196417680635853609034881, 150094038119761737476004331
Offset: 0
Programs
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Maple
exp(x/(1-3*x)^(5/3)) ; taylor(%,x=0,60) ; L := gfun[seriestolist](%) ; seq( op(i,L)*(i-1)!,i=1..nops(L)) ; # R. J. Mathar, Apr 08 2025
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PARI
a(n) = n!*sum(k=0, n, 3^(n-k)*binomial(n+2*k/3-1, n-k)/k!);
Formula
a(n) = n! * Sum_{k=0..n} 3^(n-k) * binomial(n+2*k/3-1,n-k)/k!.
a(n) ~ 2^(-3/2) * 5^(3/16) * 3^(n + 1/8) * n^(n - 3/16) * exp(-3^(-3/2)*5^(-1/4)*n^(1/4)/2 + 8*5^(-5/8)*3^(-3/4)*n^(5/8) - n). - Vaclav Kotesovec, Apr 19 2025