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A356199
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a(n) = Sum_{k=0..n} (n*k+1)^(k-1) * Stirling2(n,k).
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1
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1, 1, 6, 122, 5991, 556152, 84245291, 18956006323, 5940695613628, 2474958812797662, 1323229303771318595, 883245295259143164922, 719968321620942410875645, 703829776430361739799683993, 812798413118207226439408790038, 1094718407894086754989907938078190
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} (n*k+1)^(k-1) * Stirling2(n,k).
a(n) = [x^n] Sum_{k>=0} (n*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).
a(n) = n! * [x^n] 1/exp(LambertW((1 - exp(x))*n)/n) for n > 0, a(0) = 1.
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MAPLE
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b:= proc(n, k, m) option remember; `if`(n=0,
(k*m+1)^(m-1), m*b(n-1, k, m)+b(n-1, k, m+1))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..19);
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MATHEMATICA
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b[n_, k_, m_] := b[n, k, m] = If[n == 0,
(k*m+1)^(m-1), m*b[n-1, k, m] + b[n-1, k, m+1]];
a[n_] := b[n, n, 0];
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PROG
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(PARI) a(n) = sum(k=0, n, (n*k+1)^(k-1) * stirling(n, k, 2)); \\ Michel Marcus, Aug 04 2022
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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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