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A275207 Expansion of (A(x)^2+A(x^2))/2 where A(x) = A001006(x).

%I A275207 #17 Mar 08 2023 04:58:49
%S A275207 1,1,3,6,16,38,100,256,681,1805,4867,13162,35925,98469,271511,751656,
%T A275207 2089963,5831451,16326785,45847770,129108926,364498596,1031486590,
%U A275207 2925337352,8313215743,23668977163,67507773621,192859753310,551821400008,1581188102590
%N A275207 Expansion of (A(x)^2+A(x^2))/2 where A(x) = A001006(x).
%C A275207 Analog of A275165 with Motzkin numbers replacing connected graph counts.
%H A275207 Alois P. Heinz, <a href="/A275207/b275207.txt">Table of n, a(n) for n = 0..1000</a>
%F A275207 a(2n+1) = A275208(2n+1).
%F A275207 Conjecture: a(2n+1) = A026940(n+1).
%F A275207 Conjecture D-finite with recurrence -3*(n+4)*(n+3)*(29*n-32)*a(n) +10*(29*n-40)*(n+3)*(n+2)*a(n-1) +2*(n+1)*(149*n^2 +208*n-450)*a(n-2) -2*n*(559*n^2 -381*n-1630)*a(n-3) +4*(-68*n^3 +531*n^2 -904*n+351)*a(n-4) +2*(103*n^3-1701*n^2+5330*n -3600)*a(n-5) +18*(11*n^3 -209*n^2 +834*n -778)*a(n-6) +6*n*(269*n-830)*(n-5)*a(n-7) +9*(n-5)*(n-6)*(95*n-134)*a(n-8)=0. - _R. J. Mathar_, Mar 07 2023
%F A275207 a(n) ~ 3^(n + 5/2) / (2 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 08 2023
%p A275207 b:= proc(n) option remember; `if`(n<2, 1,
%p A275207       ((3*(n-1))*b(n-2)+(1+2*n)*b(n-1))/(n+2))
%p A275207     end:
%p A275207 a:= proc(n) option remember; add(b(j)*b(n-j), j=0..n/2)-
%p A275207       `if`(n::odd, 0, (t-> t*(t-1)/2)(b(n/2)))
%p A275207     end:
%p A275207 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jul 19 2016
%t A275207 b[n_] := b[n] = If[n<2, 1, ((3*(n-1))*b[n-2] + (1+2*n)*b[n-1])/(n+2)];
%t A275207 a[n_] := a[n] = Sum[b[j]*b[n-j], {j, 0, n/2}] - If[OddQ[n], 0, Function[t, t*(t-1)/2][b[n/2]]];
%t A275207 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, May 16 2017, after _Alois P. Heinz_ *)
%Y A275207 Cf. A275208, A026940.
%K A275207 nonn
%O A275207 0,3
%A A275207 _R. J. Mathar_, Jul 19 2016