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

0,3

Comments

Stirling transform of A001405.

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

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)).

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

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}]

Crossrefs

Cf. A001405, A005773, A064856, A305406.

Sequence in context: A352174 A124532 A343795 * A074728 A087860 A307593

Adjacent sequences: A305557 A305558 A305559 * A305561 A305562 A305563

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 21 2018

Status

approved