A242817 a(n) = B(n,n), where B(n,x) = Sum_{k=0..n} Stirling2(n,k)*x^k are the Bell polynomials (also known as exponential polynomials or Touchard polynomials).

1, 1, 6, 57, 756, 12880, 268098, 6593839, 187104200, 6016681467, 216229931110, 8588688990640, 373625770888956, 17666550789597073, 902162954264563306, 49482106424507339565, 2901159958960121863952, 181069240855214001514460, 11985869691525854175222222

Offset: 0

Author

Emanuele Munarini, May 23 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..370
• Eric Weisstein's World of Mathematics, Bell Polynomial.
• Wikipedia, Touchard polynomials

Crossrefs

Cf. A035051, A292866, A346654, A346655.

Main diagonal of A189233 and of A292860.

Programs

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}]

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

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

Extensions

Name corrected by Pontus von Brömssen, Dec 05 2020