OFFSET
0,3
COMMENTS
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.
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.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
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.
(1) A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3.
(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.
(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.
EXAMPLE
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 + ...
such that A(x) = F(x*F(x)^5) where F(x) = 1 + x*F(x)^3 begins
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 + ...
RELATED SERIES.
Let B(x) = A(x*B(x)^2) which begins
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 + ...
then
( (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 + ...
is an integer series.
PROG
(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)) ) )}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* G.f. A(x) = F(x*F(x)^5), where F(x) = 1 + x*F(x)^3 */
{a(n) = my(F = 1); for(i=1, n, F = 1 + x*F^3 + x*O(x^n));
polcoeff( subst(F, x, x*F^5), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 29 2023
STATUS
approved