A352279 a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1).
1, 2, 4, 10, 32, 114, 448, 1978, 9472, 48738, 270336, 1595114, 9965568, 65852882, 457326592, 3329243546, 25356271616, 201326396098, 1663597019136, 14279558011850, 127044810702848, 1170023757062450, 11136610150121472, 109395885009537402, 1107781178494025728, 11549900930966957346
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..576
Programs
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Mathematica
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[n - 1, 2 k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 25}] nmax = 25; CoefficientList[Series[Exp[2 Sinh[x]], {x, 0, nmax}], x] Range[0, nmax]! Table[Sum[(-1)^k * Binomial[n, k] * BellB[k, -1] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 27 2022 *) -
PARI
my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sinh(x)))) \\ Seiichi Manyama, Mar 26 2022
Formula
E.g.f.: exp( 2 * sinh(x) ).
a(n) = Sum_{k=0..n} 2^k * A136630(n,k). - Seiichi Manyama, Feb 18 2025