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A289066 Recurrence a(n+2) = Sum_{k=0..n} binomial(n,k)a(k)a(n+1-k) with a(0)=3, a(1)=1. 15
3, 1, 3, 10, 39, 184, 1047, 7000, 53571, 460936, 4404603, 46296040, 530878719, 6595091944, 88232942847, 1264741738120, 19337532032091, 314144393039176, 5403576523773603, 98110258621524520, 1875097757416854999, 37629001852534817704, 791088129700026499047 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

One of a family of integer sequences whose e.g.f.s satisfy the differential equation f''(z) = f'(z)f(z). For more details, see A289064.

LINKS

Stanislav Sykora, Table of n, a(n) for n = 0..200

S. Sykora, Sequences related to the differential equation f'' = af'f, Stan's Library, Vol. VI, Jun 2017.

FORMULA

E.g.f.: -sqrt(7)/tanh(z*sqrt(7)/2 - acosh(3/sqrt(2))).

E.g.f. for the sequence (-1)^(n+1)*a(n): -sqrt(7)/tanh(z*sqrt(7)/2 + acosh(3/sqrt(2))).

a(n) ~ 2 * n! * 7^((n+1)/2) / log(8 + 3*sqrt(7))^(n+1). - Vaclav Kotesovec, Jun 24 2017

MAPLE

f:= proc(n) option remember; add(binomial(n-2, k)*procname(k)*procname(n-1-k), k=0..n-2) end proc:

f(0):= 3: f(1):= 1:

map(f, [$0..50]); # Robert Israel, Jul 20 2017

MATHEMATICA

a[n_] := a[n] = Sum[Binomial[n-2, k]*a[k]*a[n-k-1], {k, 0, n-2}]; a[0] = 3; a[1] = 1; Array[a, 23, 0] (* Jean-Fran├žois Alcover, Jul 20 2017 *)

PROG

(PARI) c0=3; c1=1; nmax = 200; a = vector(nmax+1); a[1]=c0; a[2]=c1; for(m=0, #a-3, a[m+3]=sum(k=0, m, binomial(m, k)*a[k+1]*a[m+2-k])); a

CROSSREFS

Sequences for other starting pairs: A000111 (1,1), A289064 (1,-1), A289065 (2,-1), A289067 (3,-1), A289068 (1,-2), A289069 (3,-2), A289070 (0,3).

Sequence in context: A170860 A170845 A025238 * A126970 A204134 A233168

Adjacent sequences:  A289063 A289064 A289065 * A289067 A289068 A289069

KEYWORD

nonn

AUTHOR

Stanislav Sykora, Jun 23 2017

STATUS

approved

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Last modified September 26 10:52 EDT 2017. Contains 292518 sequences.