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

1, 2, 10, 77, 799, 10427, 163967, 3017562, 63625324, 1512354975, 40012800675, 1166271373797, 37134022033885, 1282405154139046, 47745103281852282, 1906411492286148245, 81267367663098939459, 3683790958912910588623, 176937226305157687076779

Offset: 0

Author

Vladeta Jovovic, Feb 04 2008

Keywords

• easy
• nonn

Comments

Main diagonal of array "The first r-Bell numbers", p.3 of Mezo, A108087. - Jonathan Vos Post, Sep 25 2009

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

Links

• Robert Israel, Table of n, a(n) for n = 0..385
• R. Jakimczuk, Successive Derivatives and Integer Sequences, J. Int. Seq. 14 (2011) # 11.7.3.
• Istvan Mezo, The r-Bell numbers, arXiv:0909.4417 [math.CO], 2009-2010.
• I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1.

Crossrefs

Main diagonal of A108087.

Cf. A000110.

Programs

Maple

with(combinat): a:= n-> add(binomial(n, k)*n^(n-k)*bell(k), k=0..n):
seq(a(n), n=0..20); # Emeric Deutsch, Mar 02 2008
# Alternate:
g:= proc(n) local S;
  S:= series(exp(exp(x)+n*x-1),x,n+1);
n!*coeff(S,x,n);
end proc:
map(g, [$0..30]); # Robert Israel, Sep 29 2017
# third Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1,
      k*b(n-1, k)+ b(n-1, k+1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021

Mathematica

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 *)
Join[{1}, Table[Sum[Binomial[n,k]*n^(n-k)*BellB[k], {k,0,n}], {n,1,20}]] (* Vaclav Kotesovec, Jun 09 2020 *)

Sage

def A134980(n):
    return add(binomial(n, k)*n^(n-k)*bell_number(k) for k in (0..n))
[A134980(n) for n in (0..18)]  # Peter Luschny, May 05 2013

Formula

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

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

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000110(n+k). - Vladeta Jovovic, Nov 08 2009

a(n) = n! * [x^n] exp(exp(x) + n*x - 1). - Ilya Gutkovskiy, Sep 26 2017

a(n) ~ exp(exp(1) - 1) * n^n. - Vaclav Kotesovec, Jun 09 2020

Extensions

More terms from Emeric Deutsch, Mar 02 2008