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A187653
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Binomial cumulative sums of the central Stirling numbers of the second kind (A007820).
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0
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1, 2, 10, 115, 2108, 52006, 1606229, 59550709, 2575966264, 127343893378, 7081926869746, 437585883729512, 29740614295527535, 2205002457135885616, 177099066222770055407, 15317784128757306540986, 1419476705128570400447376
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OFFSET
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0,2
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LINKS
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Table of n, a(n) for n=0..16.
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FORMULA
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a(n) = sum(binomial(n,k)S(2k,k)),k=0..n)
a(n) ~ exp(c*(2-c)/4) * StirlingS2(2*n,n) ~ 2^(2*n-1/2)*n^(n-1/2)/(sqrt(Pi*(1-c))*exp(n-c*(2-c)/4)*(c*(2-c))^n), where c = - LambertW(-2/exp(2)) = 0.406375739959959907676958... - Vaclav Kotesovec, Jan 02 2013
O.g.f.: Sum_{n>=0} n^(2*n)/n! * x^n/(1-x)^(n+1) * exp(-n^2*x/(1-x)). - Paul D. Hanna, Jan 02 2013
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MAPLE
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seq(sum(binomial(n, k)*combinat[stirling2](2*k, k), k=0..n), n=0..12);
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MATHEMATICA
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Table[Sum[Binomial[n, k]StirlingS2[2k, k], {k, 0, n}], {n, 0, 16}]
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PROG
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(Maxima) makelist(sum(binomial(n, k)*stirling2(2*k, k), k, 0, n), n, 0, 12);
(PARI) {a(n)=polcoeff(sum(m=0, n, m^(2*m)/m!*x^m/(1-x)^(m+1)*exp(-m^2*x/(1-x+x*O(x^n)))), n)} \\ Paul D. Hanna, Jan 02 2013
for(n=0, 20, print1(a(n), ", "))
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CROSSREFS
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Cf. A007820.
Sequence in context: A113089 A054928 A132522 * A131811 A006121 A110951
Adjacent sequences: A187650 A187651 A187652 * A187654 A187655 A187656
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KEYWORD
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nonn,easy
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AUTHOR
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Emanuele Munarini, Mar 12 2011
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STATUS
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approved
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