A091046 Stirling transform of first differences of Bell numbers (A005493), if offset zero: a(n) = Sum_{k=1..n} A008277(n,k)*A005493(k).

1, 4, 20, 119, 817, 6338, 54707, 519184, 5366097, 59934937, 718748131, 9203953921, 125268224954, 1804750726306, 27426230051634, 438260834123607, 7343677070172330, 128716143768613600, 2354633702684629141, 44865189679858465163, 888784065003104357924

Offset: 1

Author

Karol A. Penson, Dec 15 2003

Keywords

• nonn

Comments

Equals A039810 * [1,2,3,...], i.e., the square of the Stirling2 triangle and the natural number vector. - Gary W. Adamson, Jan 31 2008

From Mark Wildon, Nov 01 2022: (Start)

a(n) is the number of pairs (P, P') where P' is a set partition of {1,...,n}, P is a set partition of {1,...,P} refining P, and one part of P' is distinguished.

For example, for n=2 the 4 set partition pairs for n=2 are ({{1,2}},{{1,2}*}), ({{1},{2}},{{1,2}}*), ({{1},{2}},{{1}*,{2}}), ({{1},{2}},{{1},{2}*}), where the distinguished part of the coarser partition is marked *

a(n) is the inner product in the character ring of the symmetric group S_{mn} of the characters pi^n and phi_n Ind_{S_m wr S_n}^{S_{mn}}, where pi(g) = |Fix g| is the permutation character of the natural representation of S_{mn} and phi_n is the character of the wreath product S_m wr S_n obtained by inflating the character chi^{(n-1,1)} of S_n to S_m wr S_n. (End)

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..478

Crossrefs

Cf. A000258, A005493, A039810.

Programs

Maple

s:= proc(n) option remember; expand(`if`(n=0, 1,
      x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
    end:
S:= proc(n, k) option remember; coeff(s(n), x, k) end:
b:= proc(n, k) option remember; `if`(k=0, n,
      add(S(n, j)*b(j, k-1), j=0..n))
    end:
a:= n-> b(n, 2):
seq(a(n), n=1..23);  # Alois P. Heinz, Aug 24 2021

Mathematica

len = 23;
Array[StirlingS2, {len, len}].Differences[Array[BellB, len+1]] (* Jean-François Alcover, Apr 25 2022 *)

Formula

E.g.f.: (exp(exp(x)-1)-1)*exp(exp(exp(x)-1)-1).

Representation as an infinite sum (Dobinski-type relation): a(n) = exp(exp(-1)-1)*Sum(p^n*((Sum((Stirling2(p+1, k) - Stirling2(p, k))*exp(-k), k=1..p) + exp(-(p+1)))/p!), p>=1), n = 1, 2, ....