A363111 Expansion of g.f. A(x) = F(x*F(x)^7), where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.

1, 1, 11, 127, 1547, 19652, 258069, 3481034, 47999915, 674086924, 9612919156, 138878011335, 2028718584989, 29918897595468, 444889269572286, 6663228661354420, 100430376524360459, 1522215623202615036, 23187346871707554564, 354783440893854307244

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.

Links

Table of n, a(n) for n=0..19.

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

Cf. A363304, A363308, A363309, A002293.

Sequence in context: A084975 A065543 A015598 * A181012 A168327 A282042

Adjacent sequences: A363108 A363109 A363110 * A363112 A363113 A363114

Keyword

nonn

Author

Paul D. Hanna, May 30 2023

Status

approved