A075506 Shifts one place left under 7th-order binomial transform.

1, 1, 8, 71, 729, 8842, 125399, 2026249, 36458010, 719866701, 15453821461, 358100141148, 8899677678109, 235877034446341, 6634976621814472, 197269776623577659, 6177654735731310917, 203136983117907790890, 6994626418539177737803, 251584328242318030774781

Offset: 0

Author

Wolfdieter Lang, Oct 02 2002

Keywords

• nonn
• easy
• eigen

Comments

Previous name was: Row sums of triangle A075502 (for n>=1).

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..110

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->7^(n-m)*Stirling2(n,m))); # Muniru A Asiru, Mar 20 2018

Maple

[seq(factorial(k)*coeftayl(exp((exp(7*x)-1)/7), x = 0, k), k=0..20)]; # Muniru A Asiru, Mar 20 2018

Mathematica

Table[7^n BellB[n, 1/7], {n, 0, 20}]

Formula

a(n) = sum((7^(n-m))*S2(n,m), m=0..n), with S2(n,m) = A008277(n,m) (Stirling2).

E.g.f.: exp((exp(7*x)-1)/7).

O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 7*j*x). - Ilya Gutkovskiy, Mar 20 2018

a(n) ~ 7^n * n^n * exp(n/LambertW(7*n) - 1/7 - n) / (sqrt(1 + LambertW(7*n)) * LambertW(7*n)^n). - Vaclav Kotesovec, Jul 15 2021

Extensions

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