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a(0) = 1; a(n) = 2 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1).
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%I #15 Jun 27 2022 04:16:07

%S 1,2,4,10,32,114,448,1978,9472,48738,270336,1595114,9965568,65852882,

%T 457326592,3329243546,25356271616,201326396098,1663597019136,

%U 14279558011850,127044810702848,1170023757062450,11136610150121472,109395885009537402,1107781178494025728,11549900930966957346

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

%H Seiichi Manyama, <a href="/A352279/b352279.txt">Table of n, a(n) for n = 0..576</a>

%F E.g.f.: exp( 2 * sinh(x) ).

%t 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}]

%t nmax = 25; CoefficientList[Series[Exp[2 Sinh[x]], {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[(-1)^k * Binomial[n, k] * BellB[k, -1] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 27 2022 *)

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*sinh(x)))) \\ _Seiichi Manyama_, Mar 26 2022

%Y Cf. A000557, A000807, A001861, A003724, A352280.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Mar 10 2022