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A134980 a(n) = Sum_{k=0..n} binomial(n,k)*n^(n-k)*A000110(k).

%I A134980 #51 Jan 11 2022 17:33:48
%S A134980 1,2,10,77,799,10427,163967,3017562,63625324,1512354975,40012800675,
%T A134980 1166271373797,37134022033885,1282405154139046,47745103281852282,
%U A134980 1906411492286148245,81267367663098939459,3683790958912910588623,176937226305157687076779
%N A134980 a(n) = Sum_{k=0..n} binomial(n,k)*n^(n-k)*A000110(k).
%C A134980 Main diagonal of array "The first r-Bell numbers", p.3 of Mezo, A108087. - _Jonathan Vos Post_, Sep 25 2009
%C A134980 Number of partitions of [2n] where at least n blocks contain their own index element.  a(2) = 10: 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. - _Alois P. Heinz_, Jan 07 2022
%H A134980 Robert Israel, <a href="/A134980/b134980.txt">Table of n, a(n) for n = 0..385</a>
%H A134980 R. Jakimczuk, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Jakimczuk2/jakimczuk17.html">Successive Derivatives and Integer Sequences</a>, J. Int. Seq. 14 (2011) # 11.7.3.
%H A134980 Istvan Mezo, <a href="http://arxiv.org/abs/0909.4417">The r-Bell numbers</a>, arXiv:0909.4417 [math.CO], 2009-2010.
%H A134980 I. Mezo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Mezo/mezo9.html">The r-Bell numbers</a>, J. Int. Seq. 14 (2011) # 11.1.1.
%F A134980 a(n) = exp(-1)*Sum_{k>=0} (n+k)^n/k!.
%F A134980 E.g.f.: A(x) = exp(-1)*Sum_{k>=0} (1+k*x)^k/k!.
%F A134980 a(n) = Sum_{k=0..n} Stirling1(n,k)*A000110(n+k). - _Vladeta Jovovic_, Nov 08 2009
%F A134980 a(n) = n! * [x^n] exp(exp(x) + n*x - 1). - _Ilya Gutkovskiy_, Sep 26 2017
%F A134980 a(n) ~ exp(exp(1) - 1) * n^n. - _Vaclav Kotesovec_, Jun 09 2020
%p A134980 with(combinat): a:= n-> add(binomial(n, k)*n^(n-k)*bell(k), k=0..n):
%p A134980 seq(a(n), n=0..20); # _Emeric Deutsch_, Mar 02 2008
%p A134980 # Alternate:
%p A134980 g:= proc(n) local S;
%p A134980   S:= series(exp(exp(x)+n*x-1),x,n+1);
%p A134980 n!*coeff(S,x,n);
%p A134980 end proc:
%p A134980 map(g, [$0..30]); # _Robert Israel_, Sep 29 2017
%p A134980 # third Maple program:
%p A134980 b:= proc(n, k) option remember; `if`(n=0, 1,
%p A134980       k*b(n-1, k)+ b(n-1, k+1))
%p A134980     end:
%p A134980 a:= n-> b(n$2):
%p A134980 seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 04 2021
%t A134980 a[n_] := n!*SeriesCoefficient[Exp[Exp[x] + n*x - 1], {x, 0, n}]; Array[a, 19, 0] (* _Jean-François Alcover_, Sep 28 2017, after _Ilya Gutkovskiy_ *)
%t A134980 Join[{1}, Table[Sum[Binomial[n,k]*n^(n-k)*BellB[k], {k,0,n}], {n,1,20}]] (* _Vaclav Kotesovec_, Jun 09 2020 *)
%o A134980 (Sage)
%o A134980 def A134980(n):
%o A134980     return add(binomial(n, k)*n^(n-k)*bell_number(k) for k in (0..n))
%o A134980 [A134980(n) for n in (0..18)]  # _Peter Luschny_, May 05 2013
%Y A134980 Main diagonal of A108087.
%Y A134980 Cf. A000110.
%K A134980 easy,nonn
%O A134980 0,2
%A A134980 _Vladeta Jovovic_, Feb 04 2008
%E A134980 More terms from _Emeric Deutsch_, Mar 02 2008