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

%I #17 Mar 08 2023 04:58:49

%S 1,1,3,6,16,38,100,256,681,1805,4867,13162,35925,98469,271511,751656,

%T 2089963,5831451,16326785,45847770,129108926,364498596,1031486590,

%U 2925337352,8313215743,23668977163,67507773621,192859753310,551821400008,1581188102590

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

%C Analog of A275165 with Motzkin numbers replacing connected graph counts.

%H Alois P. Heinz, <a href="/A275207/b275207.txt">Table of n, a(n) for n = 0..1000</a>

%F a(2n+1) = A275208(2n+1).

%F Conjecture: a(2n+1) = A026940(n+1).

%F 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 a(n) ~ 3^(n + 5/2) / (2 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Mar 08 2023

%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=0..n/2)-

%p `if`(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, 0, n/2}] - If[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. A275208, A026940.

%K nonn

%O 0,3

%A _R. J. Mathar_, Jul 19 2016