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

0,3

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

Cf. A035051, A292866, A346654, A346655.

Main diagonal of A189233 and of A292860.

Sequence in context: A305276 A032119 A294511 * A295238 A256016 A361291

Adjacent sequences: A242814 A242815 A242816 * A242818 A242819 A242820

Keyword

nonn

Author

Emanuele Munarini, May 23 2014

Extensions

Name corrected by Pontus von Brömssen, Dec 05 2020

Status

approved