%I #11 Oct 20 2016 02:38:04
%S 1,1,4,7,18,39,95,232,606,1663,4839,14807,47330,156611,532308,1846622,
%T 6507103,23210020,83590477,303425693,1108650850,4073443378,
%U 15039391464,55763147423,207543422052,775082175863,2903508757053,10907257755616
%N Expansion of 1/((1-x^2*c(x))(1-x-2x^2)) where c(x) is the g.f. of A000108.
%C Diagonal sums of the Jacobsthal-Catalan triangle A139377.
%H G. C. Greubel, <a href="/A135582/b135582.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = Sum_{k=0..n} J(k+1)*A132364(n-k) where J(n)=A001045(n), Jacobsthal numbers.
%F Conjecture: (-n+1)*a(n) +6*(n-2)*a(n-1) +(-7*n+19)*a(n-2) +(-7*n+13)*a(n-3) +(13*n-31)*a(n-4) +2*(-n+4)*a(n-5) +4*(-2*n+5)*a(n-6)=0. - _R. J. Mathar_, Feb 23 2015
%t Jacobsthal[n_]:= (2^n - (-1)^n)/3; g[0]:= 1; g[n_] := Sum[(i/(n - i)) * Binomial[2*n - 3*i - 1, n - 2*i], {i, 0, Floor[n/2]}]; a[n_]:= Sum[Jacobsthal[k + 1]*g[n - k], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* _G. C. Greubel_, Oct 20 2016 *)
%Y Cf. A000108, A001045, A132364.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Apr 15 2008