A275208 - OEIS

%I #16 May 16 2017 08:51:32

%S 0,1,2,6,14,38,96,256,672,1805,4846,13162,35874,98469,271384,751656,

%T 2089640,5831451,16325950,45847770,129106738,364498596,1031480792,

%U 2925337352,8313200232,23668977163,67507731786,192859753310,551821286374,1581188102590

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

%C Analog of A275166 with Motzkin numbers replacing connected graph counts.

%H Alois P. Heinz, <a href="/A275208/b275208.txt">Table of n, a(n) for n = 0..1000</a>

%F a(2n+1) = A275207(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=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. A275207, A026940.

%K nonn

%O 0,3

%A _R. J. Mathar_, Jul 19 2016