A073155 Leftmost column sequence of triangle A073153.
1, 1, 4, 14, 56, 237, 1046, 4762, 22198, 105430, 508384, 2482297, 12248416, 60980875, 305955356, 1545397464, 7852100294, 40105277640, 205798130604, 1060467961508, 5485199090812, 28469067353686, 148220323891460
Offset: 0
Examples
a(3)=a(0)*[a(2)+a(1)]+[a(1)+a(0)]*[a(1)+a(0)]+[a(2)+a(1)]*a(0) =1*[4+1] + [1+1]*[1+1] + [4+1]*1 = 5 + 2*2 + 5 = 14.
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
- N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Formula
Convolution of sequence formed from sum of adjacent terms yields the original sequence without the first term:
a(n+1) = Sum_{k=0..n} [a(k) + a(k-1)] * [a(n-k) + a(n-k-1)], where a(-1)=0.
G.f.: 1/2*(1-(1-4*x*(1+x)^2)^(1/2))/x/(1+x)^2. - Vladeta Jovovic, Oct 10 2003
a(n) = Sum_{k=0..n} C(2k,n-k)*C(k). - Paul Barry, Jul 09 2006
Conjecture: (n+1)*a(n) + (-3*n+4)*a(n-1) + 2*(-6*n+7)*a(n-2) + 2*(-6*n+11)*a(n-3) + 2*(-2*n+5)*a(n-4)=0. - R. J. Mathar, Nov 26 2012
G.f. A(x) satisfies: A(x) = 1 + x * ((1 + x) * A(x))^2. - Ilya Gutkovskiy, Jul 10 2020
Extensions
More terms from Vladeta Jovovic, Oct 10 2003