A094911 -id:A094911 - cp's OEIS frontend

A094935 E.g.f.: exp(8x)/(1-8x)^(1/8).

1, 9, 89, 1073, 18321, 476473, 17484457, 813648417, 45054110369, 2872362067433, 206710159889529, 16558892507010961, 1460688620617834801, 140655075719488236057, 14678730623948132120009

Offset: 0

Philippe Deléham, Jun 18 2004

nonn

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

With[{nn=20},CoefficientList[Series[Exp[8x]/Surd[1-8x,8],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 25 2019 *)

a(n) = Sum_{k = 0..n} A046716(n, k)*8^k.
D-finite with recurrence: a(n) +(-8*n-1)*a(n-1) +64*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 15 2019
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) * 8^n / (Gamma(1/8) * exp(n-1) * n^(7/8)). - Vaclav Kotesovec, Nov 19 2021

A095176 E.g.f.: exp(9x)/(1-9x)^(1/9).

Original entry on oeis.org

1, 10, 109, 1432, 26497, 754894, 30787885, 1603546156, 99602138593, 7128277455538, 576063289419661, 51832424202980320, 5136461847251936929, 555721381650431686582, 65167921144448534609677

Offset: 0

Philippe Deléham, Jun 20 2004

nonn

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.
From Vaclav Kotesovec, Nov 19 2021: (Start)
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)).
Equivalently, a(n) ~ n! * exp(1) * k^n / (Gamma(1/k) * n^(1 - 1/k)). (End)

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

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

cp's OEIS Frontend

A094935 E.g.f.: exp(8x)/(1-8x)^(1/8).

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