A025268 - OEIS

%I #24 Nov 19 2016 04:45:48

%S 1,1,1,1,4,11,32,95,284,860,2630,8115,25242,79080,249342,790719,

%T 2520546,8072216,25961150,83814536,271538192,882527618,2876712308,

%U 9402284815,30806948110,101172278362,332965892290,1097990333320,3627433618396

%N a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5.

%H Seiichi Manyama, <a href="/A025268/b025268.txt">Table of n, a(n) for n = 1..1857</a>

%F Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example, Phi([1]) is the Catalan numbers A000108. The present sequence is Phi([1,1,1,1]). - _Gary W. Adamson_, Oct 27 2008

%F Conjecture: n*a(n) +(n+1)*a(n-1) +10*(-2*n+5)*a(n-2) +2*(2*n-9)*a(n-3) +2*(14*n-79)*a(n-4) +40*(n-7)*a(n-5)=0. - _R. J. Mathar_, Jan 25 2015

%F G.f.: 1/2 - sqrt(8*x^4+4*x^3-4*x+1)/2. - _Vaclav Kotesovec_, Jan 25 2015

%F Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 2*(2*n-9)*a(n-3) - 8*(n-6)*a(n-4). - _Vaclav Kotesovec_, Jan 25 2015

%p Phi:=proc(t,u,M) local i,a; a:=Array(0..M);

%p for i from 0 to t-1 do a[i]:=u[i+1]; od:

%p for i from t to M do a[i]:=a[i-1]+add(a[j]*a[i-1-j],j=0..i-2); od:

%p [seq(a[i],i=0..M)]; end;

%p Phi(4,[1,1,1,1],30);

%p # _N. J. A. Sloane_, Oct 29 2008

%t nmax = 30; aa = ConstantArray[0,nmax]; aa[[1]] = 1; aa[[2]] = 1; aa[[3]] = 1; aa[[4]] = 1; Do[aa[[n]] = Sum[aa[[k]]*aa[[n-k]],{k,1,n-1}],{n,5,nmax}]; aa (* _Vaclav Kotesovec_, Jan 25 2015 *)

%Y Cf. A000108, A025262.

%K nonn

%O 1,5

%A _Clark Kimberling_