cp's OEIS Frontend

A268553 Diagonal of the rational function 1/((1 - u v - u w - v w) * (1 - x y - x z - y z)).

%I A268553 #29 Nov 11 2024 22:37:11
%S A268553 1,36,8100,2822400,1200622500,572679643536,294230074634496,
%T A268553 159259227403161600,89595913068008532900,51926300783585192250000,
%U A268553 30813565377466975498995600,18639620490164944744006041600,11456409104219869032980449440000
%N A268553 Diagonal of the rational function 1/((1 - u v - u w - v w) * (1 - x y - x z - y z)).
%H A268553 Vaclav Kotesovec, <a href="/A268553/b268553.txt">Table of n, a(n) for n = 0..200</a>
%H A268553 A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015, Eq. (B.21)
%H A268553 Jacques-Arthur Weil, <a href="http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/">Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"</a>
%F A268553 Conjecture: n^4*a(n) -9*(3*n-1)^2*(3*n-2)^2*a(n-1) = 0. - _R. J. Mathar_, Mar 11 2016
%F A268553 From _Vaclav Kotesovec_, Jul 01 2016: (Start)
%F A268553 a(n) = (3*n)!^2 / (n!)^6.
%F A268553 a(n) ~ 3^(6*n+1) / (4*Pi^2*n^2).
%F A268553 (End)
%F A268553 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
%F A268553 From _Peter Bala_, Nov 11 2024: (Start)
%F A268553 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.
%F A268553 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.
%F A268553 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.
%F A268553 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)
%p A268553 A268553 := proc(n)
%p A268553     1/(1-u*v-u*w-v*w)/(1-x*y-x*z-y*z) ;
%p A268553     coeftayl(%,x=0,2*n) ;
%p A268553     coeftayl(%,y=0,2*n) ;
%p A268553     coeftayl(%,z=0,2*n) ;
%p A268553     coeftayl(%,u=0,2*n) ;
%p A268553     coeftayl(%,v=0,2*n) ;
%p A268553     coeftayl(%,w=0,2*n) ;
%p A268553 end proc:
%p A268553 seq(A268553(n),n=0..40) ; # _R. J. Mathar_, Mar 11 2016
%t A268553 Table[(3*n)!^2 / n!^6, {n, 0, 15}] (* _Vaclav Kotesovec_, Jul 01 2016 *)
%t A268553 CoefficientList[Series[HypergeometricPFQ[{1/3, 1/3, 2/3, 2/3}, {1, 1, 1}, 729 x], {x, 0, 20}], x] (* _Benedict W. J. Irwin_, Oct 20 2016 *)
%Y A268553 Cf. A268552.
%K A268553 nonn
%O A268553 0,2
%A A268553 _N. J. A. Sloane_, Feb 29 2016