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A305404
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Expansion of Sum_{k>=0} (2*k - 1)!!*x^k/Product_{j=1..k} (1 - j*x).
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2
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1, 1, 4, 25, 217, 2416, 32839, 527185, 9761602, 204800551, 4801461049, 124402647370, 3529848676237, 108859319101261, 3625569585663484, 129689000146431205, 4958830249864725997, 201834650901695603296, 8712774828941647677019, 397596632650906687905565
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OFFSET
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0,3
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COMMENTS
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Stirling transform of A001147.
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LINKS
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Table of n, a(n) for n=0..19.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform
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FORMULA
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E.g.f.: 1/sqrt(3 - 2*exp(x)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*(2*k - 1)!!.
a(n) ~ sqrt(2/3) * n^n / ((log(3/2))^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Jul 01 2018
Conjecture: a(n) = Sum_{k>=0} k^n * binomial(2*k,k) / (2^k * 3^(k + 1/2)). - Diego Rattaggi, Oct 11 2020
O.g.f. conjectural: 1/(1 - x/(1 - 3*x/(1 - 3*x/(1 - 6*x/(1 - 5*x/(1 - 9*x/(1 - 7*x/(1 - ... -(2*n-1)*x/(1 - 3*n*x/(1 - ... )))))))))) - a continued fraction of Stieltjes-type. - Peter Bala, Dec 06 2020
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MATHEMATICA
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nmax = 19; CoefficientList[Series[Sum[(2 k - 1)!! x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 19; CoefficientList[Series[1/Sqrt[3 - 2 Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] (2 k - 1)!!, {k, 0, n}], {n, 0, 19}]
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CROSSREFS
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Cf. A000670, A001147, A004123, A305405.
Sequence in context: A047733 A198198 A007830 * A218826 A060911 A060912
Adjacent sequences: A305401 A305402 A305403 * A305405 A305406 A305407
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KEYWORD
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nonn,easy
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AUTHOR
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Ilya Gutkovskiy, May 31 2018
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STATUS
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approved
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