A095176 - OEIS

%I #12 Nov 19 2021 09:50:38

%S 1,10,109,1432,26497,754894,30787885,1603546156,99602138593,

%T 7128277455538,576063289419661,51832424202980320,5136461847251936929,

%U 555721381650431686582,65167921144448534609677

%N E.g.f.: exp(9x)/(1-9x)^(1/9).

%C Sum_{k = 0..n} A046716(n,k)*x^k give A000522(n), A081367(n), A094822(n), A094856(n), A094869(n), A094905(n), A094911(n), A094935(n) for x = 1, 2, 3, 4, 5, 6, 7, 8 respectively.

%C From _Vaclav Kotesovec_, Nov 19 2021: (Start)

%C In general, for k > 0, if e.g.f. = exp(k*x) / (1 - k*x)^(1/k), then a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * k^n / (Gamma(1/k) * exp(n-1) * n^(1 - 1/k)).

%C Equivalently, a(n) ~ n! * exp(1) * k^n / (Gamma(1/k) * n^(1 - 1/k)). (End)

%F a(n) = Sum_{k = 0..n} A046716(n, k)*9^k.

%F D-finite with recurrence a(n) +(-9*n-1)*a(n-1) +81*(n-1)*a(n-2)=0. - _R. J. Mathar_, Aug 20 2021

%F a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * 9^n / (Gamma(1/9) * exp(n-1) * n^(8/9)). - _Vaclav Kotesovec_, Nov 19 2021

%t With[{nn=20},CoefficientList[Series[Exp[9x]/Surd[1-9x,9],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 25 2020 *)

%K nonn

%O 0,2

%A _Philippe Deléham_, Jun 20 2004