A110508 - OEIS

%I #19 Aug 29 2017 16:22:10

%S 1,1,4,17,87,490,2945,18517,120340,802005,5451651,37652546,263480357,

%T 1864065017,13311094644,95816113129,694511157535,5064818563258,

%U 37135165923801,273581694291309,2024194855052180,15034769479254861

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

%C Diagonal sums of A110506.

%H G. C. Greubel, <a href="/A110508/b110508.txt">Table of n, a(n) for n = 0..1000</a> (terms n=0..200 from Vincenzo Librandi)

%F 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).

%F 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

%F a(n) ~ 9 * 2^(3*n+4) / (529 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 08 2014

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

%o (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

%K nonn,easy

%O 0,3

%A _Paul Barry_, Jul 24 2005