A351937 Expansion of e.g.f. exp( (sinh(x) + x*cosh(x)) / 2 ).
1, 1, 1, 3, 9, 24, 99, 418, 1769, 9320, 49541, 278912, 1764825, 11319784, 77850287, 570610472, 4290387409, 34316005632, 285335249065, 2455224885440, 22165590003849, 206191758121856, 1989511661589435, 19903718061574144, 204795484665487865, 2179948112062667392
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..568
Programs
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Mathematica
nmax = 25; CoefficientList[Series[Exp[(Sinh[x] + x Cosh[x])/2], {x, 0, nmax}], x] Range[0, nmax]! a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, 2 k] (k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}] -
PARI
my(x='x+O('x^30)); Vec(serlaplace(exp((sinh(x) + x*cosh(x))/2))) \\ Michel Marcus, Feb 26 2022
Formula
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * (k+1) * a(n-2*k-1).