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 A134980 a(n) = Sum_{k=0..n} binomial(n,k)*n^(n-k)*A000110(k). 3

%I

%S 1,2,10,77,799,10427,163967,3017562,63625324,1512354975,40012800675,

%T 1166271373797,37134022033885,1282405154139046,47745103281852282,

%U 1906411492286148245,81267367663098939459,3683790958912910588623,176937226305157687076779

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

%C Main diagonal of array "The first r-Bell numbers" from r=0 and from n=0, p.3 of Mezo, A108087. First 7 rows and columns of array shown. [_Jonathan Vos Post_, Sep 25 2009]

%H Robert Israel, <a href="/A134980/b134980.txt">Table of n, a(n) for n = 0..385</a>

%H 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 Istvan Mezo, <a href="http://arxiv.org/abs/0909.4417">The r-Bell numbers</a>, arXiv:0909.4417 [math.CO], 2009-2010.

%H 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 a(n) = exp(-1)*Sum_{k>=0} (n+k)^n/k!.

%F E.g.f.: A(x) = exp(-1)*Sum_{k>=0} (1+k*x)^k/k!.

%F a(n) = Sum_{k=0..n} Stirling1(n,k)*A000110(n+k). [_Vladeta Jovovic_, Nov 08 2009]

%F a(n) = n! * [x^n] exp(exp(x) + n*x - 1). - _Ilya Gutkovskiy_, Sep 26 2017

%p with(combinat): a := n -> add(binomial(n, k)*n^(n-k)*bell(k), k=0..n):

%p 1, seq(a(n), n=1..20); # _Emeric Deutsch_, Mar 02 2008

%p # Alternate:

%p g:= proc(n) local S;

%p S:= series(exp(exp(x)+n*x-1),x,n+1);

%p n!*coeff(S,x,n);

%p end proc:

%p map(g, [\$0..30]); # _Robert Israel_, Sep 29 2017

%t 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_ *)

%o (Sage)

%o def A134980(n):

%o return add(binomial(n, k)*n^(n-k)*bell_number(k) for k in (0..n))

%o [A134980(n) for n in (0..18)] # _Peter Luschny_, May 05 2013

%Y Cf. A108087.

%K easy,nonn

%O 0,2

%A _Vladeta Jovovic_, Feb 04 2008

%E More terms from _Emeric Deutsch_, Mar 02 2008

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