A289067 Recurrence a(n+2) = Sum_{k=0..n} binomial(n,k)*a(k)*a(n+1-k) with a(0)=3, a(1)=-1.

3, -1, -3, -8, -15, 14, 357, 2302, 7725, -23626, -655383, -6082538, -26422935, 192117134, 5645490477, 65726212222, 317363920005, -4755023055706, -146987610294063, -1994869987891418, -9440043721651455, 280432883707929854, 9053536431109958997, 136677605454588278542

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(11)*tanh(z*sqrt(11)/2 - arccosh(sqrt(11/2))).

E.g.f. for the sequence (-1)^(n+1)*a(n): -sqrt(11)*tanh(z*sqrt(11)/2 + arccosh(sqrt(11/2))).

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, 24, 0] (* Jean-François 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), A289066 (3,1), A289068 (1,-2), A289069 (3,-2), A289070 (0,3).

Sequence in context: A068958 A238106 A087000 * A010282 A119265 A143453

Adjacent sequences: A289064 A289065 A289066 * A289068 A289069 A289070

Keyword

sign

Author

Stanislav Sykora, Jun 23 2017

Status

approved