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A317170 Expansion of e.g.f. exp(exp(x) - 1)*BesselI(1,2*(exp(x) - 1))/(exp(x) - 1).

%I A317170 #8 Feb 16 2025 08:33:56
%S A317170 1,1,3,11,48,242,1374,8619,58923,434595,3431263,28817120,256100717,
%T A317170 2397920319,23567078396,242343368931,2600148486462,29036252825090,
%U A317170 336754427112094,4048299252733563,50357053778129599,647129716643654763,8579133975080008700,117178742009906802080,1646975673395621229201
%N A317170 Expansion of e.g.f. exp(exp(x) - 1)*BesselI(1,2*(exp(x) - 1))/(exp(x) - 1).
%C A317170 Stirling transform of the Motzkin numbers (A001006).
%H A317170 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A317170 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingTransform.html">Stirling Transform</a>
%F A317170 a(n) = Sum_{k=0..n} Stirling2(n,k)*A001006(k).
%p A317170 a:=series(exp(exp(x) - 1)*BesselI(1,2*(exp(x) - 1))/(exp(x) - 1), x=0, 26): seq(n!*coeff(a, x, n), n=0..24); # _Paolo P. Lava_, Mar 26 2019
%t A317170 nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] BesselI[1, 2 (Exp[x] - 1)]/(Exp[x] - 1), {x, 0, nmax}], x] Range[0, nmax]!
%t A317170 Table[Sum[StirlingS2[n, k] Hypergeometric2F1[(1 - k)/2, -k/2, 2, 4], {k, 0, n}], {n, 0, 24}]
%Y A317170 Cf. A001006, A064856, A086672, A317169.
%K A317170 nonn
%O A317170 0,3
%A A317170 _Ilya Gutkovskiy_, Jul 23 2018