A110508 Expansion of 1/(1-(x+x^2)c(2x)), c(x) the g.f. of A000108.

1, 1, 4, 17, 87, 490, 2945, 18517, 120340, 802005, 5451651, 37652546, 263480357, 1864065017, 13311094644, 95816113129, 694511157535, 5064818563258, 37135165923801, 273581694291309, 2024194855052180, 15034769479254861

Offset: 0

Author

Paul Barry, Jul 24 2005

Keywords

• nonn
• easy

Comments

Diagonal sums of A110506.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms n=0..200 from Vincenzo Librandi)

Programs

Mathematica

CoefficientList[Series[1/(1-(x+x^2)*(1-Sqrt[1-8*x])/(4*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

PARI

x='x+O('x^50); Vec(1/(1-(x+x^2)*(1-sqrt(1-8*x))/(4*x))) \\ G. C. Greubel, Aug 29 2017

Formula

a(0)=1; for n>0, a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..(n-k)} j*C(2n-2k-j-1, n-k-j)*C(j, k)*2^(n-k-j)/(n-k).

Conjecture: n*(3*n-7)*a(n) -4*(3*n-4)*(2*n-5)*a(n-1) +2*n*(3*n-7) +(-45*n^2+177*n-160)*a(n-3) -4*(3*n-4)*(2*n-5)*a(n-4) = 0. - R. J. Mathar, Nov 15 2011

a(n) ~ 9 * 2^(3*n+4) / (529 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 08 2014