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A091046
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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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1
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1, 4, 20, 119, 817, 6338, 54707, 519184, 5366097, 59934937, 718748131, 9203953921, 125268224954, 1804750726306, 27426230051634, 438260834123607, 7343677070172330, 128716143768613600, 2354633702684629141, 44865189679858465163, 888784065003104357924
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OFFSET
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1,2
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COMMENTS
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Equals A039810 * [1,2,3,...], i.e., the square of the Stirling2 triangle and the natural number vector. - Gary W. Adamson, Jan 31 2008
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)
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LINKS
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FORMULA
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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, ....
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MAPLE
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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):
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MATHEMATICA
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len = 23;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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