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%I #12 May 16 2017 08:51:53
%S 0,0,0,2,5,17,45,129,349,970,2658,7364,20363,56634,157750,441084,
%T 1236173,3474672,9789568,27648486,78254719,221951037,630717569,
%U 1795576937,5120472435,14625574662,41837913310,119851980508,343798008165,987445317761,2839518208661
%N Expansion of (A(x)^2-A(x^2))/2 where A(x) = A001006(x)-1.
%C Analog of A216785 with Motzkin numbers replacing connected graph counts.
%H Alois P. Heinz, <a href="/A275210/b275210.txt">Table of n, a(n) for n = 0..1000</a>
%F a(2n+1) = A275209(2n+1).
%p b:= proc(n) option remember; `if`(n<2, 1,
%p ((3*(n-1))*b(n-2)+(1+2*n)*b(n-1))/(n+2))
%p end:
%p a:= proc(n) option remember; add(b(j)*b(n-j), j=1..n/2)-
%p `if`(n=0 or n::odd, 0, (t->t*(t+1)/2)(b(n/2)))
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jul 19 2016
%t b[n_] := b[n] = If[n<2, 1, ((3*(n-1))*b[n-2] + (1+2*n)*b[n-1])/(n+2)];
%t a[n_] := a[n] = Sum[b[j]*b[n - j], {j, 1, n/2}] - If[n == 0 || OddQ[n], 0, Function[t, t*(t + 1)/2][b[n/2]]];
%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, May 16 2017, after _Alois P. Heinz_ *)
%Y Cf. A275209.
%K nonn
%O 0,4
%A _R. J. Mathar_, Jul 19 2016
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