A025268 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5, with initial values 1,1,1,1.

1, 1, 1, 1, 4, 11, 32, 95, 284, 860, 2630, 8115, 25242, 79080, 249342, 790719, 2520546, 8072216, 25961150, 83814536, 271538192, 882527618, 2876712308, 9402284815, 30806948110, 101172278362, 332965892290, 1097990333320, 3627433618396

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1857

Crossrefs

Cf. A000108, A025262.

Programs

Maple

Phi:=proc(t,u,M) local i,a; a:=Array(0..M);
for i from 0 to t-1 do a[i]:=u[i+1]; od:
for i from t to M do a[i]:=a[i-1]+add(a[j]*a[i-1-j],j=0..i-2); od:
[seq(a[i],i=0..M)]; end;
Phi(4,[1,1,1,1],30);
# N. J. A. Sloane, Oct 29 2008

Mathematica

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 *)

Formula

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

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

G.f.: 1/2 - sqrt(8*x^4+4*x^3-4*x+1)/2. - Vaclav Kotesovec, Jan 25 2015

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