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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

%I

%S 2,-1,-2,-3,0,24,108,162,-1440,-14256,-54432,177552,4432320,31796064,

%T 10520928,-2531636208,-31078494720,-119133110016,2180339762688,

%U 46923057637632,368154762301440,-2077357560938496,-101408182152625152,-1314869775259580928,-1225663306833715200

%N Recurrence a(n+2) = Sum_{k=0..n} binomial(n,k)a(k)a(n+1-k) with a(0)=2, 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="/A289065/b289065.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(6)*tanh(z*sqrt(6)/2 - acosh(sqrt(3))).

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

%t 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-François Alcover_, Jul 20 2017 *)

%o (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

%Y 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).

%K sign

%O 0,1

%A _Stanislav Sykora_, Jun 23 2017

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