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%I #11 Feb 10 2019 17:08:39
%S 1,1,1,-1,4,-21,133,-981,8244,-77694,811194,-9292075,115843000,
%T -1561272571,22618147199,-350481556959,5784147674772,-101284047800632,
%U 1875504207906184,-36616289396963678,751702523788615816,-16187581390548113842,364861626149143519378,-8590429045711448354359
%N Expansion of e.g.f. BesselI(0,2*log(1 + x)) + BesselI(1,2*log(1 + x)).
%H Robert Israel, <a href="/A306335/b306335.txt">Table of n, a(n) for n = 0..400</a>
%F a(n) = Sum_{k=0..n} Stirling1(n,k)*A001405(k).
%p E:= BesselI(0,2*log(1 + x)) + BesselI(1,2*log(1 + x)):
%p S:= series(E,x,51):
%p seq(coeff(S,x,j)*j!,j=0..50); # _Robert Israel_, Feb 10 2019
%t nmax = 23; CoefficientList[Series[BesselI[0, 2 Log[1 + x]] + BesselI[1, 2 Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[StirlingS1[n, k] Binomial[k, Floor[k/2]], {k, 0, n}], {n, 0, 23}]
%o (PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(k, k\2)); \\ _Michel Marcus_, Feb 09 2019
%Y Cf. A001405, A048994, A086672, A305560.
%K sign
%O 0,5
%A _Ilya Gutkovskiy_, Feb 08 2019