A305560 Expansion of Sum_{k>=0} binomial(k,floor(k/2))*x^k/Product_{j=1..k} (1 - j*x).
1, 1, 3, 10, 39, 176, 893, 4985, 30229, 197452, 1379655, 10250087, 80558195, 666916238, 5795111845, 52691973136, 499969246647, 4938724595994, 50679201983653, 539209298355565, 5938139329609621, 67582179415195986, 793755139140445707, 9608367683839952732, 119730171975510540577
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..550
- N. J. A. Sloane, Transforms
- Eric Weisstein's World of Mathematics, Stirling Transform
Programs
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Maple
a:= n-> add(binomial(j, floor(j/2))*Stirling2(n, j), j=0..n): seq(a(n), n=0..30); # Alois P. Heinz, Jun 21 2018
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Mathematica
nmax = 24; CoefficientList[Series[Sum[Binomial[k, Floor[k/2]] x^k/Product[1 - j x, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x] nmax = 24; CoefficientList[Series[BesselI[0, 2 (Exp[x] - 1)] + BesselI[1, 2 (Exp[x] - 1)], {x, 0, nmax}], x] Range[0, nmax]! Table[Sum[StirlingS2[n, k] Binomial[k, Floor[k/2]], {k, 0, n}], {n, 0, 24}]
Formula
E.g.f.: BesselI(0,2*(exp(x) - 1)) + BesselI(1,2*(exp(x) - 1)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*binomial(k,floor(k/2)).
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