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A305560 Expansion of Sum_{k>=0} binomial(k,floor(k/2))*x^k/Product_{j=1..k} (1 - j*x). 2

%I

%S 1,1,3,10,39,176,893,4985,30229,197452,1379655,10250087,80558195,

%T 666916238,5795111845,52691973136,499969246647,4938724595994,

%U 50679201983653,539209298355565,5938139329609621,67582179415195986,793755139140445707,9608367683839952732,119730171975510540577

%N Expansion of Sum_{k>=0} binomial(k,floor(k/2))*x^k/Product_{j=1..k} (1 - j*x).

%C Stirling transform of A001405.

%H Vaclav Kotesovec, <a href="/A305560/b305560.txt">Table of n, a(n) for n = 0..550</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>

%F E.g.f.: BesselI(0,2*(exp(x) - 1)) + BesselI(1,2*(exp(x) - 1)).

%F a(n) = Sum_{k=0..n} Stirling2(n,k)*binomial(k,floor(k/2)).

%p a:= n-> add(binomial(j, floor(j/2))*Stirling2(n, j), j=0..n):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Jun 21 2018

%t 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]

%t nmax = 24; CoefficientList[Series[BesselI[0, 2 (Exp[x] - 1)] + BesselI[1, 2 (Exp[x] - 1)], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[StirlingS2[n, k] Binomial[k, Floor[k/2]], {k, 0, n}], {n, 0, 24}]

%Y Cf. A001405, A005773, A064856, A305406.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jun 21 2018

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Last modified August 18 08:57 EDT 2019. Contains 326077 sequences. (Running on oeis4.)