A025268 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5.

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

1,5

Links

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

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

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 *)

Crossrefs

Cf. A000108, A025262.

Sequence in context: A183119 A289246 A199109 * A178520 A306419 A376071

Adjacent sequences: A025265 A025266 A025267 * A025269 A025270 A025271

Keyword

nonn

Author

Clark Kimberling

Status

approved