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 A289070 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. 15
 3, 9, 108, 2754, 120528, 8059824, 764365248, 97582435344, 16135857600768, 3354823392632064, 856584985953881088, 263495061361859433984, 96111473403635977310208, 41016996175782988022575104, 20247499012863186836834992128, 11447373157054380028382302439424 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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). 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. LINKS Stanislav Sykora, Table of n, a(n) for n = 1..100 S. Sykora, Sequences related to the differential equation f'' = af'f, Stan's Library, Vol. VI, Jun 2017. FORMULA E.g.f.: odd terms of sqrt(6)*tan(z*sqrt(3/2)). E.g.f. for (-1)^(n)*a(n): odd terms of -sqrt(6)*tanh(z*sqrt(3/2)). a(n) ~ (2n-1)! * 2^(n+2) * 3^n / Pi^(2*n). - Vaclav Kotesovec, Jun 24 2017 PROG (PARI) c0=0; c1=3; nmax = 200;   s=vector(nmax+1)); s[1]=c0; s[2]=c1;   for(m=0, #s-3, s[m+3]=sum(k=0, m, binomial(m, k)*s[k+1]*s[m+2-k]));   a = vector((nmax+1)\2, i, s[2*i]) CROSSREFS 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). Sequence in context: A125652 A232701 A018746 * A053914 A018757 A281070 Adjacent sequences:  A289067 A289068 A289069 * A289071 A289072 A289073 KEYWORD nonn AUTHOR Stanislav Sykora, Jun 23 2017 STATUS approved

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