cp's OEIS Frontend

A355347 a(n) = binomial(3*n+3,n) + binomial(3*n+2,n-1) for n >= 0.

%I A355347 #7 Jul 27 2022 10:30:58
%S A355347 1,7,44,275,1729,10948,69768,447051,2877875,18599295,120609840,
%T A355347 784384692,5114119724,33417386200,218786861392,1434903854139,
%U A355347 9425348845815,61997934676405,408323057257500,2692322893972635,17770644483690945,117406930477134480,776363580147660960
%N A355347 a(n) = binomial(3*n+3,n) + binomial(3*n+2,n-1) for n >= 0.
%C A355347 A355345(2*n*(n+1)) = (-1)^n * a(n) for n >= 1.
%C A355347 Limit_{n->oo} a(n)/a(n+1) = 4/27.
%F A355347 G.f.: G(x)^3 * (1 + x*G(x)^2) / (1 - 3*x*G(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
%F A355347 G.f.: G'(x) * (1 + x*G(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.
%F A355347 a(n) = [x^n] (1+x)/(1-x)^(2*n+4) for n >= 0.
%e A355347 G.f.: A(x) = 1 + 7*x + 44*x^2 + 275*x^3 + 1729*x^4 + 10948*x^5 + 69768*x^6 + 447051*x^7 + 2877875*x^8 + 18599295*x^9 + 120609840*x^10 + ...
%e A355347 such that
%e A355347 A(x) = G(x)^3 * (1 + x*G(x)^2) / (1 - 3*x*G(x)^2)
%e A355347 where G(x) = 1 + x*G(x)^3 begins
%e A355347 G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 + 43263*x^8 + 246675*x^9 + 1430715*x^10 + ... + A001764(n)*x^n + ...
%o A355347 (PARI) {a(n) = binomial(3*n+3,n) + binomial(3*n+2,n-1)}
%o A355347 for(n=0,22,print1(a(n),", "))
%Y A355347 Cf. A001764, A355345.
%K A355347 nonn
%O A355347 0,2
%A A355347 _Paul D. Hanna_, Jul 25 2022