OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..370
Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials
FORMULA
E.g.f.: x*f'(x)/f(x), where f(x) is the generating series for sequence A035051.
a(n) ~ (exp(1/LambertW(1)-2)/LambertW(1))^n * n^n / sqrt(1+LambertW(1)). - Vaclav Kotesovec, May 23 2014
Conjecture: It appears that the equation a(x)*e^x = Sum_{n=0..oo} ( (n^x*x^n)/n! ) is true for every positive integer x. - Nicolas Nagel, Apr 20 2016 [This is just the special case k=x of the formula B(k,x) = e^(-x) * Sum_{n=0..oo} n^k*x^n/n!; see for example the World of Mathematics link. - Pontus von Brömssen, Dec 05 2020]
a(n) = n! * [x^n] exp(n*(exp(x)-1)). - Alois P. Heinz, May 17 2016
a(n) = [x^n] Sum_{k=0..n} n^k*x^k/Product_{j=1..k} (1 - j*x). - Ilya Gutkovskiy, May 31 2018
MAPLE
A:= proc(n, k) option remember; `if`(n=0, 1, (1+
add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
end:
a:= n-> A(n$2):
seq(a(n), n=0..20); # Alois P. Heinz, May 17 2016
MATHEMATICA
Table[BellB[n, n], {n, 0, 100}]
PROG
(Maxima) a(n):=stirling2(n, 0)+sum(stirling2(n, k)*n^k, k, 1, n);
makelist(a(n), n, 0, 30);
(PARI) a(n) = sum(k=0, n, stirling(n, k, 2)*n^k); \\ Michel Marcus, Apr 20 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
Emanuele Munarini, May 23 2014
EXTENSIONS
Name corrected by Pontus von Brömssen, Dec 05 2020
STATUS
approved