OFFSET
0,3
COMMENTS
Compare the g.f. A(x) = F(x*F(x)^7) to F(-x*F(x)^7) = 1/F(x), where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.
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 A002293.
(1) A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4.
(2) A(x) = B(x/A(x)^3) where B(x) = A(x*B(x)^3) = F( x*B(x)^3 * F(x*B(x)^3)^7 ).
(3) a(n) = Sum_{k=1..n} 7*k* binomial(4*k+1, k) * binomial(4*n+3*k, n-k) / ((4*k+1)*(4*n+3*k)) for n > 0, with a(0) = 1.
EXAMPLE
G.f.: A(x) = 1 + x + 11*x^2 + 127*x^3 + 1547*x^4 + 19652*x^5 + 258069*x^6 + 3481034*x^7 + 47999915*x^8 + 674086924*x^9 + ...
such that A(x) = F(x*F(x)^7) where F(x) = 1 + x*F(x)^4 begins
F(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 + 53820*x^7 + ... + A002293(n)*x^n + ...
RELATED SERIES.
Let B(x) = A(x*B(x)^3) = ( Series_Reversion( x/A(x)^3 )/x )^(1/3) which begins
B(x) = 1 + x + 14*x^2 + 238*x^3 + 4578*x^4 + 95130*x^5 + 2082150*x^6 + 47295990*x^7 + 1104598378*x^8 + ...
then
( (B(x) - 1)/x )^(1/7) = 1 + 2*x + 22*x^2 + 350*x^3 + 6538*x^4 + 133658*x^5 + 2895214*x^6 + 65294502*x^7 + ... + A363304(n)*x^n + ...
is an integer series.
PROG
(PARI) {a(n) = if(n==0, 1, sum(k=1, n, 7*k* binomial(4*k+1, k) * binomial(4*n+3*k, n-k) / ((4*k+1)*(4*n+3*k)) ) )}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* G.f. A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4 */
{a(n) = my(F = 1); for(i=1, n, F = 1 + x*F^4 + x*O(x^n));
polcoeff( subst(F, x, x*F^7), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 30 2023
STATUS
approved