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

1, 1, 3, 6, 16, 38, 100, 256, 681, 1805, 4867, 13162, 35925, 98469, 271511, 751656, 2089963, 5831451, 16326785, 45847770, 129108926, 364498596, 1031486590, 2925337352, 8313215743, 23668977163, 67507773621, 192859753310, 551821400008, 1581188102590

Offset

0,3

Comments

Analog of A275165 with Motzkin numbers replacing connected graph counts.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

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

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

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

a(n) ~ 3^(n + 5/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023

Maple

b:= proc(n) option remember; `if`(n<2, 1,
      ((3*(n-1))*b(n-2)+(1+2*n)*b(n-1))/(n+2))
    end:
a:= proc(n) option remember; add(b(j)*b(n-j), j=0..n/2)-
      `if`(n::odd, 0, (t-> t*(t-1)/2)(b(n/2)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 19 2016

Mathematica

b[n_] := b[n] = If[n<2, 1, ((3*(n-1))*b[n-2] + (1+2*n)*b[n-1])/(n+2)];
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]]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 16 2017, after Alois P. Heinz *)

Crossrefs

Cf. A275208, A026940.

Sequence in context: A190735 A096588 A256943 * A073079 A143560 A279685

Adjacent sequences: A275204 A275205 A275206 * A275208 A275209 A275210

Keyword

nonn

Author

R. J. Mathar, Jul 19 2016

Status

approved