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A363309 Expansion of g.f. A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.

%I A363309 #19 May 30 2023 11:50:49
%S A363309 1,1,8,67,590,5403,51034,494268,4886794,49153835,501631980,5182767291,
%T A363309 54115252508,570206217940,6055948422280,64765311313944,
%U A363309 696876526961130,7539151412082315,81957518070961472,894826829565106185,9808173152466891270,107888887505651377475
%N A363309 Expansion of g.f. A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.
%C A363309 Compare the g.f. A(x) = F(x*F(x)^5) to F(-x*F(x)^5) = 1/F(x), where F(x) = 1 + x*F(x)^3 is the g.f. of A001764.
%C A363309 Conjecture: given A(x) = F(x*F(x)^(2*n-1)) where F(x) = 1 + x*F(x)^n, let B(x) = A(x*B(x)^(n-1)), then ((B(x) - 1)/x)^(1/(2*n-1)) is an integer series for n >= 1. Incidentally, the function A(x) = F(x*F(x)^(2*n-1)) is interesting because F(-x*F(x)^(2*n-1)) = 1/F(x) when F(x) = 1 + x*F(x)^n. This sequence illustrates the case for n = 3; for n = 2, see A363308.
%H A363309 Paul D. Hanna, <a href="/A363309/b363309.txt">Table of n, a(n) for n = 0..300</a>
%F A363309 G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined as follows; here, F(x) is the g.f. of A001764.
%F A363309 (1) A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3.
%F A363309 (2) A(x) = B(x/A(x)^2) where B(x) = A(x*B(x)^2) = F( x*B(x)^2 * F(x*B(x)^2)^5 ) is the g.f. of A363310.
%F A363309 (3) a(n) = Sum_{k=1..n} 5*k* binomial(3*k+1, k) * binomial(3*n+2*k, n-k) / ((3*k+1)*(3*n+2*k)) for n > 0, with a(0) = 1.
%e A363309 G.f.: A(x) = 1 + x + 8*x^2 + 67*x^3 + 590*x^4 + 5403*x^5 + 51034*x^6 + 494268*x^7 + 4886794*x^8 + 49153835*x^9 + 501631980*x^10 + ...
%e A363309 such that A(x) = F(x*F(x)^5) where F(x) = 1 + x*F(x)^3 begins
%e A363309 F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 + ... + A001764(n)*x^n + ...
%e A363309 RELATED SERIES.
%e A363309 Let B(x) = A(x*B(x)^2) which begins
%e A363309 B(x) = 1 + x + 10*x^2 + 120*x^3 + 1620*x^4 + 23560*x^5 + 360352*x^6 + 5714800*x^7 + 93129840*x^8 + ... + A363310(n)*x^n + ...
%e A363309 then
%e A363309 ( (B(x) - 1)/x )^(1/5) = 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 A363309 is an integer series.
%o A363309 (PARI) {a(n) = if(n==0, 1, sum(k=1, n, 5*k* binomial(3*k+1, k) * binomial(3*n+2*k, n-k) / ((3*k+1)*(3*n+2*k)) ) )}
%o A363309 for(n=0, 30, print1(a(n), ", "))
%o A363309 (PARI) /* G.f. A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3 */
%o A363309 {a(n) = my(F = 1); for(i=1,n, F = 1 + x*F^3 + x*O(x^n));
%o A363309 polcoeff( subst(F, x, x*F^5), n)}
%o A363309 for(n=0, 30, print1(a(n), ", "))
%Y A363309 Cf.  A363308, A363309, A363310, A363311, A363111, A001764.
%K A363309 nonn
%O A363309 0,3
%A A363309 _Paul D. Hanna_, May 29 2023