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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).

%I A242817 #50 Jul 27 2021 04:32:37
%S A242817 1,1,6,57,756,12880,268098,6593839,187104200,6016681467,216229931110,
%T A242817 8588688990640,373625770888956,17666550789597073,902162954264563306,
%U A242817 49482106424507339565,2901159958960121863952,181069240855214001514460,11985869691525854175222222
%N 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).
%H A242817 Alois P. Heinz, <a href="/A242817/b242817.txt">Table of n, a(n) for n = 0..370</a>
%H A242817 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%H A242817 Wikipedia, <a href="https://en.wikipedia.org/wiki/Touchard_polynomials">Touchard polynomials</a>
%F A242817 E.g.f.: x*f'(x)/f(x), where f(x) is the generating series for sequence A035051.
%F A242817 a(n) ~ (exp(1/LambertW(1)-2)/LambertW(1))^n * n^n / sqrt(1+LambertW(1)). - _Vaclav Kotesovec_, May 23 2014
%F A242817 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]
%F A242817 a(n) = n! * [x^n] exp(n*(exp(x)-1)). - _Alois P. Heinz_, May 17 2016
%F A242817 a(n) = [x^n] Sum_{k=0..n} n^k*x^k/Product_{j=1..k} (1 - j*x). - _Ilya Gutkovskiy_, May 31 2018
%p A242817 A:= proc(n, k) option remember; `if`(n=0, 1, (1+
%p A242817       add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
%p A242817     end:
%p A242817 a:= n-> A(n$2):
%p A242817 seq(a(n), n=0..20);  # _Alois P. Heinz_, May 17 2016
%t A242817 Table[BellB[n, n], {n, 0, 100}]
%o A242817 (Maxima) a(n):=stirling2(n,0)+sum(stirling2(n,k)*n^k,k,1,n);
%o A242817 makelist(a(n),n,0,30);
%o A242817 (PARI) a(n) = sum(k=0, n, stirling(n,k,2)*n^k); \\ _Michel Marcus_, Apr 20 2016
%Y A242817 Cf. A035051, A292866, A346654, A346655.
%Y A242817 Main diagonal of A189233 and of A292860.
%K A242817 nonn
%O A242817 0,3
%A A242817 _Emanuele Munarini_, May 23 2014
%E A242817 Name corrected by _Pontus von Brömssen_, Dec 05 2020