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A061685
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Generalized Bell numbers.
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4
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1, 1, 9, 298, 25097, 4383626, 1394519922, 738298190981, 608765840524809, 742996254490626106, 1289282092211451157634, 3078466688415490018129781, 9844321075186192301310239858, 41209705023068976933023104392293, 221473347301087557264532943397984133
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listen;
history;
text;
internal format)
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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_{pi} n!/(k(1)! * 1!^k(1) * k(2)! * 2!^k(2) * ... * k(n)! * n!^k(n)) * (n!/(1!^k(1) * 2!^k(2) * ... * n!^k(n)))^L, where pi runs through all partitions k(1) + 2 * k( 2) + ... + n * k(n) = n, with L = 3.
a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k)^4 * (n-k) * a(k). - Ilya Gutkovskiy, Jul 12 2020
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MAPLE
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b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-i)*binomial(n-1, i-1)/i!^3, i=1..n))
end:
a:= n-> b(n)*n!^3:
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MATHEMATICA
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b[n_] := b[n] = If[n==0, 1, Sum[b[n-i]*Binomial[n-1, i-1]/i!^3, {i, n}]];
a[n_] := b[n]*n!^3;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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