A351892 Expansion of e.g.f. exp( sinh(sqrt(3)*x) / sqrt(3) ).

1, 1, 1, 4, 13, 40, 205, 952, 4921, 31168, 189145, 1318528, 9843781, 74869888, 632536933, 5475991552, 49996774897, 485393809408, 4829958877105, 50858117779456, 554544498995965, 6259096187060224, 73822470722135293, 894846287081242624, 11261265009125680681, 146272258394568687616

Offset: 0

Author

Ilya Gutkovskiy, Feb 24 2022

Keywords

• nonn

Crossrefs

Cf. A003724, A009229, A136630, A247452, A351891, A381277.

Programs

Mathematica

nmax = 25; CoefficientList[Series[Exp[Sinh[Sqrt[3] x]/Sqrt[3]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] 3^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) * 3^k * a(n-2*k-1).

a(n) = Sum_{k=0..n} 3^((n-k)/2) * A136630(n,k). - Seiichi Manyama, Feb 20 2025