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A309084
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a(n) = exp(3) * Sum_{k>=0} (-3)^k*k^n/k!.
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4
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1, -3, 6, -3, -21, 24, 195, -111, -3072, -4053, 57003, 277854, -697539, -12261567, -29861778, 371727465, 3511027599, 2028432480, -188521156857, -1470389129931, 1655487186864, 121873222577823, 915525253963023, -2095901567014530, -103715912230195863, -836215492271268459
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f.: Sum_{j>=0} (-3)^j*x^j / Product_{k=1..j} (1 - k*x).
E.g.f.: exp(3*(1 - exp(x))).
a(n) = Sum_{k=0..n} (-3)^k * Stirling2(n,k).
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MAPLE
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b:= proc(n, m) option remember; `if`(n=0,
(-3)^m, m*b(n-1, m)+b(n-1, m+1))
end:
a:= n-> b(n, 0):
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MATHEMATICA
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Table[Exp[3] Sum[(-3)^k k^n/k!, {k, 0, Infinity}], {n, 0, 25}]
Table[BellB[n, -3], {n, 0, 25}]
nmax = 25; CoefficientList[Series[Sum[(-3)^j x^j/Product[(1 - k x), {k, 1, j}] , {j, 0, nmax}], {x, 0, nmax}], x]
nmax = 25; CoefficientList[Series[Exp[3 (1 - Exp[x])], {x, 0, nmax}], x] Range[0, nmax]!
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PROG
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(Magma) [1] cat [(&+[((-3)^k*StirlingSecond(m, k)):k in [0..m]]):m in [1..25]]; // Marius A. Burtea, Jul 27 2019
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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