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

0,3

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

Example

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.

Crossrefs

Cf. A073153, A073156, A073157.

Cf. A052709, A360076, A360082, A360083.

Sequence in context: A132837 A259808 A149491 * A365114 A370720 A346816

Adjacent sequences: A073152 A073153 A073154 * A073156 A073157 A073158

Keyword

easy,nonn

Author

Paul D. Hanna, Jul 29 2002

Extensions

More terms from Vladeta Jovovic, Oct 10 2003

Status

approved