A378694 G.f. A(x) satisfies A(x) = 1 + x*A(x)^7/(1 - x*A(x)^6).

1, 1, 8, 91, 1210, 17577, 270314, 4326070, 71300386, 1202012254, 20630488004, 359279348424, 6332747550808, 112761701957119, 2025325557546780, 36650364776763804, 667570101840389826, 12229542931765845994, 225183117821591853440, 4165207037639796288385

Offset

0,3

Links

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

Formula

G.f. A(x) satisfies A(x) = 1/(1 - x*A(x)^6/(1 - x*A(x)^6)).

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) / (1 - x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(n+(s-1)*k-1,n-k)/(t*k+u*(n-k)+r).

From Seiichi Manyama, Sep 29 2020: (Start)

G.f. A(x) satisfies: A(x) = 1 - x * A(x)^6 * (1 - 2 * A(x)).

a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(6*n+k+1,n)/(6*n+k+1).

a(n) = ( (-1)^n / (6*n+1) ) * Sum_{k=0..n} (-2)^(n-k) * binomial(6*n+1,k) * binomial(7*n-k,n-k).

a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(7*n-k,n-1-k) for n > 0.

G.f.: 1 + Series_Reversion( x / ((1+x)^6 * (1+2*x)) ). (End)

Prog

(PARI) a(n, r=1, s=1, t=7, u=6) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(n+(s-1)*k-1, n-k)/(t*k+u*(n-k)+r));

Crossrefs

Column k=6 of A336573.

Cf. A378685, A378688, A378692.

Sequence in context: A184709 A190943 A180912 * A234280 A386722 A298013

Adjacent sequences: A378691 A378692 A378693 * A378695 A378696 A378697

Keyword

nonn

Author

Seiichi Manyama, Dec 04 2024

Status

approved