A075507 Shifts one place left under 8th-order binomial transform.

1, 1, 9, 89, 1009, 13457, 210105, 3747753, 74565473, 1628999841, 38704241897, 993034281593, 27340167242321, 803154583649329, 25050853217628313, 826165199464341705, 28707262835597618369, 1047731789671001235265, 40053733152627299592137, 1599910554128824794493593

Offset

0,3

Comments

Previous name was: Row sums of triangle A075503 (for n>=1).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..422 (first 111 terms from Muniru A Asiru)

Formula

a(n) = Sum_{m=0..n} 8^(n-m)*S2(n,m), with S2(n,m) = A008277(n,m) (Stirling2).

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

O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 8*j*x). - Ilya Gutkovskiy, Mar 20 2018

a(n) ~ 8^n * n^n * exp(n/LambertW(8*n) - 1/8 - n) / (sqrt(1 + LambertW(8*n)) * LambertW(8*n)^n). - Vaclav Kotesovec, Jul 15 2021

Maple

[seq(factorial(k)*coeftayl(exp((exp(8*x)-1)/8), x = 0, k), k=0..20)]; # Muniru A Asiru, Mar 20 2018
# Alternative:
b:= proc(n, m) option remember;
     `if`(n=0, 1, 8*m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..22);  # Alois P. Heinz, Jun 24 2026

Mathematica

Table[8^n BellB[n, 1/8], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

Prog

(GAP) List([0..20], n->Sum([0..n], m->8^(n-m)*Stirling2(n, m))); # Muniru A Asiru, Mar 20 2018

Crossrefs

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

Sequence in context: A199759 A069573 A101679 * A094935 A258388 A296618

Adjacent sequences: A075504 A075505 A075506 * A075508 A075509 A075510

Keyword

nonn,easy,eigen

Author

Wolfdieter Lang, Oct 02 2002

Extensions

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

Status

approved