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%I #20 Jun 23 2023 07:47:08
%S 1,42,13590,7410480,5074665750,3931541906292,3290234596753104,
%T 2903719469665734720,2664470557292509315350,2519171054960424071707500,
%U 2438942560726150616825027340,2407205497290665533295978551680,2414129024898590207617757303701200
%N Diagonal of the rational function 1/((1 - u v - u w - v w - u v w) * (1 - x y - x z - y z)).
%C Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r. - _Peter Bala_, Jun 22 2023
%H Vaclav Kotesovec, <a href="/A268552/b268552.txt">Table of n, a(n) for n = 0..200</a>
%H A. Bostan, S. Boukraa, J.-M. Maillard, 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, 2015
%H 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 Recurrence: (n-1)^2*n^4*(2*n - 1)^2*(3504*n^5 - 30952*n^4 + 107572*n^3 - 183680*n^2 + 154013*n - 50763)*a(n) = 3*(n-1)^2*(3*n - 2)*(3*n - 1)*(798912*n^9 - 8654880*n^8 + 39805024*n^7 - 101548488*n^6 + 157599040*n^5 - 153473292*n^4 + 93212371*n^3 - 33837996*n^2 + 6644187*n - 547020)*a(n-1) - 9*(3*n - 5)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(7989120*n^9 - 94537920*n^8 + 478555808*n^7 - 1351912112*n^6 + 2330100416*n^5 - 2513561312*n^4 + 1671938288*n^3 - 648448700*n^2 + 130707255*n - 10800900)*a(n-2) + 27*(2*n - 5)^2*(3*n - 8)*(3*n - 7)*(3*n - 5)*(3*n - 4)*(3*n - 2)*(3*n - 1)*(3504*n^5 - 13432*n^4 + 18804*n^3 - 11636*n^2 + 3081*n - 306)*a(n-3). - _Vaclav Kotesovec_, Jul 01 2016
%F a(n) ~ c * d^n / (Pi^2 * n^2), where d = 1189.7580084904576415418942340231454997... is the root of the equation -19683 + 415530*d - 1539*d^2 + d^3 = 0 and c = 0.44151111077974450880059816263885416848395811427... is the root of the equation -1 + 72*c - 384*c^2 + 512*c^3 = 0. - _Vaclav Kotesovec_, Jul 01 2016
%p A268552 := proc(n)
%p 1/(1-u*v-u*w-v*w-u*v*w)/(1-x*y-x*z-y*z) ;
%p coeftayl(%,x=0,n) ;
%p coeftayl(%,y=0,n) ;
%p coeftayl(%,z=0,n) ;
%p coeftayl(%,u=0,n) ;
%p coeftayl(%,v=0,n) ;
%p coeftayl(%,w=0,n) ;
%p end proc:
%p seq(A268552(2*n),n=0..40) ; # _R. J. Mathar_, Mar 10 2016
%t f = 1/((1 - u v - u w - v w - u v w)*(1 - x y - x z - y z));
%t a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}]&, f, {x, y, z, u, v, w}];
%t Table[a[2n], {n, 0, 20}] (* _Jean-François Alcover_, Dec 03 2017 *)
%Y Cf. A268545 - A268555.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Feb 29 2016
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