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A289065 Recurrence a(n+2) = Sum_{k=0..n} binomial(n,k)a(k)a(n+1-k) with a(0)=2, a(1)=-1. 16
2, -1, -2, -3, 0, 24, 108, 162, -1440, -14256, -54432, 177552, 4432320, 31796064, 10520928, -2531636208, -31078494720, -119133110016, 2180339762688, 46923057637632, 368154762301440, -2077357560938496, -101408182152625152, -1314869775259580928, -1225663306833715200 (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(6)*tanh(z*sqrt(6)/2 - acosh(sqrt(3))).

E.g.f. for the sequence (-1)^(n+1)*a(n): -sqrt(6)*tanh(z*sqrt(6)/2 + acosh(sqrt(3))).

MATHEMATICA

a[n_] := a[n] = Sum[Binomial[n-2, k]*a[k]*a[n-k-1], {k, 0, n-2}]; a[0] = 2; a[1] = -1; Array[a, 25, 0] (* Jean-Fran├žois Alcover, Jul 20 2017 *)

PROG

(PARI) c0=2; 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), A289066 (3,1), A289067 (3,-1), A289068 (1,-2), A289069 (3,-2), A289070 (0,3).

Sequence in context: A249858 A249857 A249777 * A132815 A167684 A094646

Adjacent sequences:  A289062 A289063 A289064 * A289066 A289067 A289068

KEYWORD

sign

AUTHOR

Stanislav Sykora, Jun 23 2017

STATUS

approved

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