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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.
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%I #19 May 30 2023 11:50:49

%S 1,1,8,67,590,5403,51034,494268,4886794,49153835,501631980,5182767291,

%T 54115252508,570206217940,6055948422280,64765311313944,

%U 696876526961130,7539151412082315,81957518070961472,894826829565106185,9808173152466891270,107888887505651377475

%N 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 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 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 Paul D. Hanna, <a href="/A363309/b363309.txt">Table of n, a(n) for n = 0..300</a>

%F 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 (1) A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3.

%F (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 (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 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 such that A(x) = F(x*F(x)^5) where F(x) = 1 + x*F(x)^3 begins

%e 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 RELATED SERIES.

%e Let B(x) = A(x*B(x)^2) which begins

%e 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 then

%e ( (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 is an integer series.

%o (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 for(n=0, 30, print1(a(n), ", "))

%o (PARI) /* G.f. A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3 */

%o {a(n) = my(F = 1); for(i=1,n, F = 1 + x*F^3 + x*O(x^n));

%o polcoeff( subst(F, x, x*F^5), n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A363308, A363309, A363310, A363311, A363111, A001764.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 29 2023