A323325 Coefficients a(n) of x^n*y^n*z^n in function A = A(x,y,z) such that A = 1 + x*B*C, B = 1 + y*C*A, and C = 1 + z*A*B, for n >= 0.

1, 3, 54, 1500, 51450, 2000376, 84523824, 3789772272, 177645575250, 8619977394600, 429995817814992, 21940447761898848, 1140809521021467024, 60271001744583000000, 3228065652622114800000, 174953984892890573016000, 9580981919014895332205250, 529506723911785149640077000, 29503054954798945147262250000, 1655865088781640962375927700000, 93546915429941104997312052606000, 5316303408243224817579669328104000

Offset: 0

Author

Paul D. Hanna, Jan 12 2019

Keywords

• nonn

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A323324.

Programs

PARI

{a(n) = my(A=1, B=1, C=1); for(i=0, 2*n,
A = 1 + x*B*C +x*O(x^n);
B = 1 + y*A*C +y*O(y^n);
C = 1 + z*A*B +z*O(z^n));
polcoeff(polcoeff(polcoeff(A, n, x), n, y), n, z)}
for(n=0, 30, print1(a(n), ", "))

PARI

/* Using binomial formula */
{a(n) = 0^n/4 + 3/4 * binomial(2*n, n)^3 / (n+1)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = 3/4 * binomial(2*n, n)^3 / (n+1) for n >= 1, with a(0) = 1.

a(n) = 3/4 * (2*n)!^3 / (n!^6 * (n+1)) for n >= 1.