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

%I A162482 #12 Apr 04 2024 10:05:47
%S A162482 1,4,14,53,218,945,4235,19441,90947,432030,2078416,10105435,49578341,
%T A162482 245131321,1220218293,6110131376,30756858405,155547919269,
%U A162482 789965192900,4027121386190,20600180351659,105707046807196,543973305719611
%N A162482 Expansion of (1/(1-x)^3)*M(x/(1-x)^3), M(x) the g.f. of Motzkin numbers A001006.
%F A162482 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);
%F A162482 a(n) = Sum{k=0..n} C(n+2k+2,n-k)*A001006(k).
%F A162482 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
%p A162482 A162482 := proc(n)
%p A162482     add(binomial(n+2*k+2,n-k)*A001006(k),k=0..n) ;
%p A162482 end proc:
%p A162482 seq(A162482(n),n=0..40) ; # _R. J. Mathar_, Feb 10 2015
%t A162482 m[n_] := m[n] = If[n == 0, 1, m[n-1] + Sum[m[k]*m[n-2-k], {k, 0, n-2}]];
%t A162482 a[n_] := Sum[Binomial[n+2k+2, n-k]*m[k], {k, 0, n}];
%t A162482 Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Apr 04 2024 *)
%Y A162482 Cf. A001006, A162481.
%K A162482 nonn
%O A162482 0,2
%A A162482 _Paul Barry_, Jul 04 2009