A162482 Expansion of (1/(1-x)^3)*M(x/(1-x)^3), M(x) the g.f. of Motzkin numbers A001006.

1, 4, 14, 53, 218, 945, 4235, 19441, 90947, 432030, 2078416, 10105435, 49578341, 245131321, 1220218293, 6110131376, 30756858405, 155547919269, 789965192900, 4027121386190, 20600180351659, 105707046807196, 543973305719611

Offset: 0

Author

Paul Barry, Jul 04 2009

Keywords

• nonn

Crossrefs

Cf. A001006, A162481.

Programs

Maple

A162482 := proc(n)
    add(binomial(n+2*k+2,n-k)*A001006(k),k=0..n) ;
end proc:
seq(A162482(n),n=0..40) ; # R. J. Mathar, Feb 10 2015

Mathematica

m[n_] := m[n] = If[n == 0, 1, m[n-1] + Sum[m[k]*m[n-2-k], {k, 0, n-2}]];
a[n_] := Sum[Binomial[n+2k+2, n-k]*m[k], {k, 0, n}];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Apr 04 2024 *)

Formula

G.f.: 1/((1-x)^3-x-x^2/((1-x)^3-x-x^2/((1-x)^3-x-x^2/((1-x)^3-x-x^2/(1-... (continued fraction);

a(n) = Sum{k=0..n} C(n+2k+2,n-k)*A001006(k).

Conjecture: (n+2)*a(n) +4*(-2*n-1)*a(n-1) +18*(n-1)*a(n-2) +13*(-2*n+5)*a(n-3) +17*(n-4)*a(n-4) +3*(-2*n+11)*a(n-5) +(n-7)*a(n-6)=0. - R. J. Mathar, Feb 10 2015