A352280 a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1).
1, 3, 9, 30, 117, 516, 2493, 13152, 75177, 460272, 3003921, 20806176, 152114013, 1169842368, 9435180357, 79553524224, 699531782481, 6400932102912, 60820145019801, 599036357936640, 6105903392066373, 64309189153428480, 698936466350352717, 7828833281592926208, 90270159223293364473
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..565
Programs
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Mathematica
a[0] = 1; a[n_] := a[n] = 3 Sum[Binomial[n - 1, 2 k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 24}] nmax = 24; CoefficientList[Series[Exp[3 Sinh[x]], {x, 0, nmax}], x] Range[0, nmax]! -
PARI
my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(3*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022
Formula
E.g.f.: exp( 3 * sinh(x) ).
a(n) = Sum_{k=0..n} 3^k * A136630(n,k). - Seiichi Manyama, Feb 18 2025