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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
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: A364495 A340483 A018746 * A053914 A018757 A281070
KEYWORD
nonn
AUTHOR
Stanislav Sykora, Jun 23 2017
STATUS
approved

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Last modified June 16 15:23 EDT 2024. Contains 373432 sequences. (Running on oeis4.)