A384417 Expansion of g.f. cosh(9*arctanh(8*sqrt(x))).

1, 2592, 1230336, 294469632, 49690312704, 6822215811072, 818458027622400, 89312567167549440, 9086229152658358272, 875874088323041460224, 80899222450192930308096, 7217466034064795168145408, 625687045828728598806134784, 52946875811413468120885493760, 4389120887020725640048536453120

Offset: 0

Author

Karol A. Penson, May 28 2025

Keywords

• nonn

Crossrefs

Cf. A383928, A384335, A285043, A285044, A285045, A285046.

Programs

Maple

seq(coeff(series((1 + 2304*x + 516096*x^2 + 22020096*x^3 + 150994944*x^4)/(-64*x + 1)^(9/2), x, 15), x, k), k=0..14);

Mathematica

CoefficientList[Series[Cosh[9*ArcTanh[8*Sqrt[x]]],{x,0,14}],x] (* Stefano Spezia, May 29 2025 *)

Formula

a(n) = 16^n*(105 + 400*n + 3392*n^2 + 512*n^3 + 4096*n^4)*(2*n)!/(105*(n!)^2).

O.g.f.: (1 + 2304*x + 516096*x^2 + 22020096*x^3 + 150994944*x^4)/(-64*x + 1)^(9/2).

E.g.f.: exp(32*x)*((105 + 512*x*(269 + 256*x*(73 + 512*x)))*BesselI(0, 32*x) + 512*x*(25 + 256*x*(65 + 512*x))*BesselI(1, 32*x))/105 + (131072*x*hypergeom([3/2, 2, 2], [1, 1, 1], 64*x))/105.