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A363310 Expansion of g.f. A(x) satisfying A(x) = 1 + x*G(x)^5, where G(x) = 1 + x*(G(x)^3 + G(x)^5) is the g.f. of A363311.

%I A363310 #10 Jul 13 2023 08:36:55
%S A363310 1,1,10,120,1620,23560,360352,5714800,93129840,1550132320,26242225600,
%T A363310 450448137216,7821608426880,137145465358080,2424899712359680,
%U A363310 43186456105340160,774013543036174080,13949937641606981120,252666943472167541760,4596736161565468815360
%N A363310 Expansion of g.f. A(x) satisfying A(x) = 1 + x*G(x)^5, where G(x) = 1 + x*(G(x)^3 + G(x)^5) is the g.f. of A363311.
%H A363310 Seiichi Manyama, <a href="/A363310/b363310.txt">Table of n, a(n) for n = 0..500</a>
%F A363310 G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined as follows; here, B(x) is the g.f. of A363309 and F(x) is the g.f. of A001764.
%F A363310 (1) A(x) = 1 + x*G(x)^5, where G(x) = 1 + x*(G(x)^3 + G(x)^5) is the g.f. of A363311.
%F A363310 (2) A(x) = B(x*A(x)^2) where B(x) = F(x*F(x)^5) and F(x) = 1 + x*F(x)^3.
%F A363310 (3) A(x) = sqrt( (1/x)*Series_Reversion( x/B(x)^2 ) ), where B(x) is the g.f. of A363309.
%F A363310 (4) a(n) = Sum_{k=0..n-1} 5*binomial(n-1, k)*binomial(3*n+2*k+2, n-1)/(3*n+2*k+2) for n > 1 with a(0) = 1.
%e A363310 G.f.: A(x) =  = 1 + x + 10*x^2 + 120*x^3 + 1620*x^4 + 23560*x^5 + 360352*x^6 + 5714800*x^7 + 93129840*x^8 + 1550132320*x^9 + 26242225600*x^10 + ...
%e A363310 such that A(x) = 1 + x*G(x)^5 where
%e A363310 G(x) = 1 + 2*x + 16*x^2 + 180*x^3 + 2360*x^4 + 33760*x^5 + 510928*x^6 + 8043440*x^7 + ... + A363311(n)*x^n + ...
%e A363310 satisfies G(x) = 1 + x*(G(x)^3 + G(x)^5).
%e A363310 Also, A(x) = B(x*A(x)^2) where B(x) = A(x/B(x)^2) begins
%e A363310 B(x) = 1 + x + 8*x^2 + 67*x^3 + 590*x^4 + 5403*x^5 + 51034*x^6 + 494268*x^7 + ... + A363309(n)*x^n + ...
%o A363310 (PARI) {a(n) = if(n==0,1, sum(k=0, n-1, 5*binomial(n-1, k)*binomial(3*n+2*k+2, n-1)/(3*n+2*k+2) ) )}
%o A363310 for(n=0, 20, print1(a(n), ", "))
%Y A363310 Cf. A363309, A363311, A001764.
%K A363310 nonn
%O A363310 0,3
%A A363310 _Paul D. Hanna_, May 29 2023