A336589 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselI(0,2*sqrt(1 - exp(x))).
1, 0, -3, -19, -75, 574, 25795, 579963, 9342529, 21955076, -7954085799, -535479422655, -25206613635203, -871888114433454, -7465407495946777, 2538884115164554199, 344689220434285963905, 31689538033223254172648, 2273498459548301881979029
Offset: 0
Keywords
Programs
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Mathematica
nmax = 18; CoefficientList[Series[Exp[x] BesselI[0, 2 Sqrt[1 - Exp[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2 Table[n! Sum[(-1)^k StirlingS2[n + 1, k + 1]/k!, {k, 0, n}], {n, 0, 18}] -
PARI
a(n) = n! * sum(k=0, n, (-1)^k*stirling(n+1,k+1,2) / k!); \\ Michel Marcus, Jul 29 2020
Formula
a(n) = n! * Sum_{k=0..n} (-1)^k * Stirling2(n+1,k+1) / k!.