A268553 - OEIS

A268553

Diagonal of the rational function 1/((1 - u v - u w - v w) * (1 - x y - x z - y z)).

1, 36, 8100, 2822400, 1200622500, 572679643536, 294230074634496, 159259227403161600, 89595913068008532900, 51926300783585192250000, 30813565377466975498995600, 18639620490164944744006041600, 11456409104219869032980449440000

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OFFSET

0,2

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015, Eq. (B.21)

Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

FORMULA

Conjecture: n^4*a(n) -9*(3*n-1)^2*(3*n-2)^2*a(n-1) = 0. - R. J. Mathar, Mar 11 2016

From Vaclav Kotesovec, Jul 01 2016: (Start)

a(n) = (3*n)!^2 / (n!)^6.

a(n) ~ 3^(6*n+1) / (4*Pi^2*n^2).

(End)

G.f.: 4F3(1/3,1/3,2/3,2/3; 1,1,1; 729*x). Mathar's conjecture above is true. - Benedict W. J. Irwin, Oct 20 2016

From Peter Bala, Nov 11 2024: (Start)

a(n) = [x^n] F(x)^n, where F(x)^(1/36) = 1 + 7*x + 77*x^2 + 16004*x^3 + 4724082*x^4 + 1685299234*x^5 + 677278114038*x^6 + 295443291847791*x^7 + 136845776517061880*x^8 + 66356719714684604206*x^9 + 33360966330484890531781*x^10 + ... appears to have integer coefficients.

Conjecture 1. Let m be an integer. The sequence defined by u(n) = [x^n] F(x)^(m*n/36) satisfies the supercongruences u(n*p^r) === u(n*p^(r-1)) (mod p^r) for all primes p >= 5 and all positive integers n and r.

Let E(x) = exp(Sum_{n >= 1} a(n)*x^n/n). Then E(x)^(1/36) = 1 + x + 113*x^2 + 26246*x^3 + 8370174*x^4 + 3192850645*x^5 + 1366644640572*x^6 + 633922091635053*x^7 + 312001398547051724*x^8 + 160711315511105814931*x^9 + 85821749989729644162164*x^10 + ... appears to have integer coefficients.

Conjecture 2. Let m be an integer. The sequence defined by v(n) = [x^n] E(x)^(m*n/36) satisfies the supercongruences v(n*p^r) === v(n*p^(r-1)) (mod p^r) for all primes p >= 5 and all positive integers n and r. (End)

MAPLE

A268553 := proc(n)

1/(1-u*v-u*w-v*w)/(1-x*y-x*z-y*z) ;

coeftayl(%, x=0, 2*n) ;

coeftayl(%, y=0, 2*n) ;

coeftayl(%, z=0, 2*n) ;

coeftayl(%, u=0, 2*n) ;

coeftayl(%, v=0, 2*n) ;

coeftayl(%, w=0, 2*n) ;

end proc:

seq(A268553(n), n=0..40) ; # R. J. Mathar, Mar 11 2016

MATHEMATICA