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A305405
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Expansion of Sum_{k>=0} k!!*x^k/Product_{j=1..k} (1 - j*x).
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2
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1, 1, 3, 10, 41, 201, 1126, 7043, 48603, 366298, 2987189, 26163501, 244654150, 2430411335, 25539609327, 282834656434, 3290175964577, 40089424302657, 510340938343270, 6772086558823547, 93481666812344979, 1339885322519303434, 19907413622297965373, 306126204811557339045
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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E.g.f.: 1 + exp((exp(x) - 1)^2/2)*(exp(x) - 1)*(1 + sqrt(Pi/2)*erf((exp(x) - 1)/sqrt(2))).
a(n) = Sum_{k=0..n} Stirling2(n,k)*k!!.
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MAPLE
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b:= proc(n, m) option remember;
`if`(n=0, doublefactorial(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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nmax = 23; CoefficientList[Series[Sum[k!! x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 23; CoefficientList[Series[1 + Exp[(E^x - 1)^2/2] (Exp[x] - 1) (1 + Sqrt[Pi/2] Erf[(Exp[x] - 1)/Sqrt[2]]), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] k!!, {k, 0, n}], {n, 0, 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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