%I #11 Mar 26 2022 13:40:57

%S 1,3,9,30,117,516,2493,13152,75177,460272,3003921,20806176,152114013,

%T 1169842368,9435180357,79553524224,699531782481,6400932102912,

%U 60820145019801,599036357936640,6105903392066373,64309189153428480,698936466350352717,7828833281592926208,90270159223293364473

%N a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1).

%H Seiichi Manyama, <a href="/A352280/b352280.txt">Table of n, a(n) for n = 0..565</a>

%F E.g.f.: exp( 3 * sinh(x) ).

%t 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}]

%t nmax = 24; CoefficientList[Series[Exp[3 Sinh[x]], {x, 0, nmax}], x] Range[0, nmax]!

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(3*sinh(x)))) \\ _Seiichi Manyama_, Mar 26 2022

%Y Cf. A003724, A027710, A080527, A107403, A352279.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Mar 10 2022