A351891 Expansion of e.g.f. exp( sinh(sqrt(2)*x) / sqrt(2) ).

1, 1, 1, 3, 9, 25, 105, 443, 1969, 10609, 57265, 338547, 2190969, 14498185, 104277849, 784965803, 6150938593, 51229928929, 440694547681, 3967606065891, 37247506348905, 361022009762809, 3645855348771273, 38001754007842715, 409302848055407761, 4558828622414199121

Offset: 0

Author

Ilya Gutkovskiy, Feb 24 2022

Keywords

• nonn

Crossrefs

Cf. A003724, A009229, A055882, A136630, A351892, A381277.

Programs

Mathematica

nmax = 25; CoefficientList[Series[Exp[Sinh[Sqrt[2] x]/Sqrt[2]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] 2^k a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}]

Formula

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

a(n) = Sum_{k=0..n} 2^((n-k)/2) * A136630(n,k). - Seiichi Manyama, Feb 20 2025