%I #16 Nov 23 2024 05:43:53

%S 3,9,108,2754,120528,8059824,764365248,97582435344,16135857600768,

%T 3354823392632064,856584985953881088,263495061361859433984,

%U 96111473403635977310208,41016996175782988022575104,20247499012863186836834992128,11447373157054380028382302439424

%N a(n) = c(2n-1), where c(n+2) = Sum_{k=0..n} binomial(n,k)c(k)c(n+1-k) with c(0)=0, c(1)=3.

%C The sequence c(n) is one of a family of integer sequences whose e.g.f.s satisfy the differential equation f''(z) = f'(z)f(z).

%C Since c(0)=0, all its even terms are zero, and only the odd terms are listed here. For more details, see A289064 and the link.

%H Stanislav Sykora, <a href="/A289070/b289070.txt">Table of n, a(n) for n = 1..100</a>

%H Stanislav Sykora, <a href="http://dx.doi.org/10.3247/SL6Math17.001">Sequences related to the differential equation f'' = af'f</a>, Stan's Library, Vol. VI, Jun 2017.

%F E.g.f.: odd terms of sqrt(6)*tan(z*sqrt(3/2)).

%F E.g.f. for (-1)^(n)*a(n): odd terms of -sqrt(6)*tanh(z*sqrt(3/2)).

%F a(n) ~ (2n-1)! * 2^(n+2) * 3^n / Pi^(2*n). - _Vaclav Kotesovec_, Jun 24 2017

%o (PARI) c0=0;c1=3;nmax = 200;

%o s=vector(nmax+1);s[1]=c0;s[2]=c1;

%o for(m=0,#s-3,s[m+3]=sum(k=0,m,binomial(m,k)*s[k+1]*s[m+2-k]));

%o a = vector((nmax+1)\2,i,s[2*i])

%Y Sequences for other starting pairs: A000111 (1,1), A289064 (1,-1), A289065 (2,-1), A289066 (3,1), A289067 (3,-1), A289068 (1,-2), A289069 (3,-2).

%K nonn

%O 1,1

%A _Stanislav Sykora_, Jun 23 2017