A075508 Shifts one place left under 9th-order binomial transform.

1, 1, 10, 109, 1351, 19612, 333451, 6493069, 141264820, 3376695763, 87799365343, 2465959810690, 74353064138749, 2393123710957813, 81812390963020066, 2958191064076428793, 112727516544416978299, 4513118224822056822772, 189305466502867876489519

Offset: 0

Author

Wolfdieter Lang, Oct 02 2002

Keywords

• nonn
• easy
• eigen

Comments

Previous name was: Row sums of triangle A075504 (for n>=1).

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..108

Crossrefs

Shifts one place left under k-th order binomial transform, k=1..10: A000110, A004211, A004212, A004213, A005011, A005012, A075506, A075507, A075508, A075509.

Programs

GAP

List([0..20],n->Sum([0..n],m->9^(n-m)*Stirling2(n,m))); # Muniru A Asiru, Mar 20 2018

Maple

[seq(factorial(k)*coeftayl(exp((exp(9*x)-1)/9), x = 0, k), k=0..20)]; # Muniru A Asiru, Mar 20 2018

Mathematica

Table[9^n BellB[n, 1/9], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

Formula

a(n) = Sum_{m=0..n} 9^(n-m)*S2(n,m), with S2(n,m) = A008277(n,m) (Stirling2).

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

O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 9*j*x). - Ilya Gutkovskiy, Mar 20 2018

a(n) ~ 9^n * n^n * exp(n/LambertW(9*n) - 1/9 - n) / (sqrt(1 + LambertW(9*n)) * LambertW(9*n)^n). - Vaclav Kotesovec, Jul 15 2021

Extensions

a(0)=1 inserted and new name by Vladimir Reshetnikov, Oct 20 2015