A289067 - OEIS

%I #19 Aug 17 2018 19:47:14

%S 3,-1,-3,-8,-15,14,357,2302,7725,-23626,-655383,-6082538,-26422935,

%T 192117134,5645490477,65726212222,317363920005,-4755023055706,

%U -146987610294063,-1994869987891418,-9440043721651455,280432883707929854,9053536431109958997,136677605454588278542

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

%C 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.

%H Stanislav Sykora, <a href="/A289067/b289067.txt">Table of n, a(n) for n = 0..200</a>

%H S. Sykora, <a href="http://dx.doi.org/10.3247/SL6Math17.001">Sequences related to the differential equation f'' = af'f</a>, Stan's Library, Vol. VI, Jun 2017.

%F E.g.f.: -sqrt(11)*tanh(z*sqrt(11)/2 - arccosh(sqrt(11/2))).

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

%t 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 *)

%o (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

%Y 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).

%K sign

%O 0,1

%A _Stanislav Sykora_, Jun 23 2017