A352254 Expansion of e.g.f. exp( x * sinh(x) / 2 ) (even powers only).

1, 1, 5, 48, 753, 16880, 507579, 19509042, 927229553, 53126200872, 3597373129635, 283321938437318, 25614466939850169, 2629191169850594388, 303549146372282854883, 39103024746814973908890, 5581172267077778765676129, 877211696663645448333041072, 151002471269513108372760683523

Offset: 0

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

• nonn

Crossrefs

Cf. A000248, A000807, A003724, A003727, A005046, A009233, A351937, A352253.

Programs

Mathematica

nmax = 36; Take[CoefficientList[Series[Exp[x Sinh[x]/2], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n - 1, 2 k - 1] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

PARI

my(x='x+O('x^40), v=Vec(serlaplace(exp(x*sinh(x)/2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(2*n-1,2*k-1) * k * a(n-k).