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A275209 Expansion of (A(x)^2+A(x^2))/2 where A(x) = A001006(x)-1.

%I A275209 #12 May 16 2017 08:51:42
%S A275209 0,0,1,2,7,17,49,129,358,970,2679,7364,20414,56634,157877,441084,
%T A275209 1236496,3474672,9790403,27648486,78256907,221951037,630723367,
%U A275209 1795576937,5120487946,14625574662,41837955145,119851980508,343798121799,987445317761,2839518519233
%N A275209 Expansion of (A(x)^2+A(x^2))/2 where A(x) = A001006(x)-1.
%C A275209 Analog of A274934 with Motzkin numbers replacing connected graph counts.
%H A275209 Alois P. Heinz, <a href="/A275209/b275209.txt">Table of n, a(n) for n = 0..1000</a>
%F A275209 a(2n+1) = A275210(2n+1).
%p A275209 b:= proc(n) option remember; `if`(n<2, 1,
%p A275209       ((3*(n-1))*b(n-2)+(1+2*n)*b(n-1))/(n+2))
%p A275209     end:
%p A275209 a:= proc(n) option remember; add(b(j)*b(n-j), j=1..n/2)-
%p A275209       `if`(n::odd, 0, (t->t*(t-1)/2)(b(n/2)))
%p A275209     end:
%p A275209 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jul 19 2016
%t A275209 b[n_] := b[n] = If[n<2, 1, ((3*(n-1))*b[n-2] + (1+2*n)*b[n-1])/(n+2)];
%t A275209 a[n_] := a[n] = Sum[b[j]*b[n-j], {j, 1, n/2}] - If[OddQ[n], 0, Function[t, t*(t - 1)/2][b[n/2]]];
%t A275209 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, May 16 2017, after _Alois P. Heinz_ *)
%Y A275209 Cf. A275210.
%K A275209 nonn
%O A275209 0,4
%A A275209 _R. J. Mathar_, Jul 19 2016