A363304 Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^4 + A(x)^7).

1, 2, 22, 350, 6538, 133658, 2895214, 65294502, 1516963346, 36056007602, 872615973766, 21430572885422, 532737957899290, 13379121740808266, 338941379999841758, 8651415618928816886, 222278432539991439906, 5743974149517874477922, 149192980850883703986166

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..500

Formula

G.f.: A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.

(1) A(x) = 1 + x*(A(x)^4 + A(x)^7).

(2) a(n) = Sum_{k=0..n} binomial(n, k)*binomial(4*n+3*k+1, n)/(4*n+3*k+1) for n >= 0.

Example

G.f.: A(x) = 1 + 2*x + 22*x^2 + 350*x^3 + 6538*x^4 + 133658*x^5 + 2895214*x^6 + 65294502*x^7 + 1516963346*x^8 + 36056007602*x^9 + ...
where A(x) = 1 + x*(A(x)^4 + A(x)^7).
RELATED SERIES.
A(x)^4 = 1 + 8*x + 112*x^2 + 1960*x^3 + 38528*x^4 + 813064*x^5 + 17998512*x^6 + 412364968*x^7 + ...
A(x)^7 = 1 + 14*x + 238*x^2 + 4578*x^3 + 95130*x^4 + 2082150*x^5 + 47295990*x^6 + 1104598378*x^7 + ...

Prog

(PARI) {a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+3*k+1, n)/(4*n+3*k+1) )}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A027307, A363311, A363111.

Sequence in context: A364826 A245113 A363006 * A078232 A151615 A111985

Adjacent sequences: A363301 A363302 A363303 * A363305 A363306 A363307

Keyword

nonn

Author

Paul D. Hanna, May 29 2023

Status

approved