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%I #10 Feb 09 2019 07:05:24
%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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