cp's OEIS Frontend

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

%I A091046 #35 Nov 02 2022 07:49:45
%S A091046 1,4,20,119,817,6338,54707,519184,5366097,59934937,718748131,
%T A091046 9203953921,125268224954,1804750726306,27426230051634,438260834123607,
%U A091046 7343677070172330,128716143768613600,2354633702684629141,44865189679858465163,888784065003104357924
%N A091046 Stirling transform of first differences of Bell numbers (A005493), if offset zero: a(n) = Sum_{k=1..n} A008277(n,k)*A005493(k).
%C A091046 Equals A039810 * [1,2,3,...], i.e., the square of the Stirling2 triangle and the natural number vector. - _Gary W. Adamson_, Jan 31 2008
%C A091046 From _Mark Wildon_, Nov 01 2022: (Start)
%C A091046 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.
%C A091046 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 *
%C A091046 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)
%H A091046 Alois P. Heinz, <a href="/A091046/b091046.txt">Table of n, a(n) for n = 1..478</a>
%F A091046 E.g.f.: (exp(exp(x)-1)-1)*exp(exp(exp(x)-1)-1).
%F A091046 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, ....
%p A091046 s:= proc(n) option remember; expand(`if`(n=0, 1,
%p A091046       x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))
%p A091046     end:
%p A091046 S:= proc(n, k) option remember; coeff(s(n), x, k) end:
%p A091046 b:= proc(n, k) option remember; `if`(k=0, n,
%p A091046       add(S(n, j)*b(j, k-1), j=0..n))
%p A091046     end:
%p A091046 a:= n-> b(n, 2):
%p A091046 seq(a(n), n=1..23);  # _Alois P. Heinz_, Aug 24 2021
%t A091046 len = 23;
%t A091046 Array[StirlingS2, {len, len}].Differences[Array[BellB, len+1]] (* _Jean-François Alcover_, Apr 25 2022 *)
%Y A091046 Cf. A000258, A005493, A039810.
%K A091046 nonn
%O A091046 1,2
%A A091046 _Karol A. Penson_, Dec 15 2003