A025271 - OEIS

%I #21 Jan 13 2024 10:44:33

%S 1,1,2,1,6,18,52,165,518,1646,5308,17258,56604,187108,622632,2084461,

%T 7016134,23730006,80610156,274911614,940915892,3230919164,11127525464,

%U 38429281122,133052559772,461740643276,1605877668824,5596283069300

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

%H Filippo Disanto, <a href="http://arxiv.org/abs/1202.5668">The size of the biggest Caterpillar subtree in binary rooted planar trees</a>, arXiv preprint arXiv:1202.5668 [math.CO], 2012-2013. See the sequence F-(3).

%F 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

%F 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

%F Recurrence: n*a(n) = 2*(2*n-3)*a(n-1) - 16*(n-6)*a(n-4). - _Vaclav Kotesovec_, Jan 25 2015

%p For a Maple program see A214198.

%t 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 *)

%o (PARI) default(seriesprecision, 100); Vec((1-sqrt(1-4*x+16*x^4))/2 + O(x^50)) \\ _Michel Marcus_, Nov 22 2014

%K nonn

%O 1,3

%A _Clark Kimberling_