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Stirling transform of first differences of Bell numbers (A005493), if offset zero: a(n) = Sum_{k=1..n} A008277(n,k)*A005493(k).
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%I #35 Nov 02 2022 07:49:45

%S 1,4,20,119,817,6338,54707,519184,5366097,59934937,718748131,

%T 9203953921,125268224954,1804750726306,27426230051634,438260834123607,

%U 7343677070172330,128716143768613600,2354633702684629141,44865189679858465163,888784065003104357924

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

%C 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 From _Mark Wildon_, Nov 01 2022: (Start)

%C 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 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 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 Alois P. Heinz, <a href="/A091046/b091046.txt">Table of n, a(n) for n = 1..478</a>

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

%F 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 s:= proc(n) option remember; expand(`if`(n=0, 1,

%p x*add(s(n-j)*binomial(n-1, j-1), j=1..n)))

%p end:

%p S:= proc(n, k) option remember; coeff(s(n), x, k) end:

%p b:= proc(n, k) option remember; `if`(k=0, n,

%p add(S(n, j)*b(j, k-1), j=0..n))

%p end:

%p a:= n-> b(n, 2):

%p seq(a(n), n=1..23); # _Alois P. Heinz_, Aug 24 2021

%t len = 23;

%t Array[StirlingS2, {len, len}].Differences[Array[BellB, len+1]] (* _Jean-François Alcover_, Apr 25 2022 *)

%Y Cf. A000258, A005493, A039810.

%K nonn

%O 1,2

%A _Karol A. Penson_, Dec 15 2003