A025271 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 5, with initial values 1,1,2,1.

1, 1, 2, 1, 6, 18, 52, 165, 518, 1646, 5308, 17258, 56604, 187108, 622632, 2084461, 7016134, 23730006, 80610156, 274911614, 940915892, 3230919164, 11127525464, 38429281122, 133052559772, 461740643276, 1605877668824, 5596283069300

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Links

• Filippo Disanto, The size of the biggest Caterpillar subtree in binary rooted planar trees, arXiv preprint arXiv:1202.5668 [math.CO], 2012-2013. See the sequence F-(3).

Programs

Maple

For a Maple program see A214198.

Mathematica

nmax = 30; aa = ConstantArray[0,nmax]; aa[[1]] = 1; aa[[2]] = 1; aa[[3]] = 2; 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 *)

PARI

default(seriesprecision, 100); Vec((1-sqrt(1-4*x+16*x^4))/2 + O(x^50)) \\ Michel Marcus, Nov 22 2014

Formula

G.f.: (1/2)*(1-sqrt(1-4*x+2^(k+1)*x^(k+1))) with k=3. - N. J. A. Sloane, Jul 07 2012

Conjecture: n*a(n) +(n+1)*a(n-1) +(n+8)*a(n-2) +42*(-2*n+7)*a(n-3) +16*(n-6)*a(n-4) +80*(n-7)*a(n-5) +336*(n-8)*a(n-6)=0. - R. J. Mathar, Nov 21 2014

Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 16*(n-6)*a(n-4). - Vaclav Kotesovec, Jan 25 2015