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a(n) = binomial(3*n+3,n) + binomial(3*n+2,n-1) for n >= 0.
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%I #7 Jul 27 2022 10:30:58

%S 1,7,44,275,1729,10948,69768,447051,2877875,18599295,120609840,

%T 784384692,5114119724,33417386200,218786861392,1434903854139,

%U 9425348845815,61997934676405,408323057257500,2692322893972635,17770644483690945,117406930477134480,776363580147660960

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

%C A355345(2*n*(n+1)) = (-1)^n * a(n) for n >= 1.

%C Limit_{n->oo} a(n)/a(n+1) = 4/27.

%F 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 G.f.: G'(x) * (1 + x*G(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

%F a(n) = [x^n] (1+x)/(1-x)^(2*n+4) for n >= 0.

%e 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 such that

%e A(x) = G(x)^3 * (1 + x*G(x)^2) / (1 - 3*x*G(x)^2)

%e where G(x) = 1 + x*G(x)^3 begins

%e 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 (PARI) {a(n) = binomial(3*n+3,n) + binomial(3*n+2,n-1)}

%o for(n=0,22,print1(a(n),", "))

%Y Cf. A001764, A355345.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 25 2022