A268552 Diagonal of the rational function 1/((1 - u v - u w - v w - u v w) * (1 - x y - x z - y z)).

1, 42, 13590, 7410480, 5074665750, 3931541906292, 3290234596753104, 2903719469665734720, 2664470557292509315350, 2519171054960424071707500, 2438942560726150616825027340, 2407205497290665533295978551680, 2414129024898590207617757303701200

Offset

0,2

Comments

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

Links

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

A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227, 2015

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

Formula

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

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

Maple

A268552 := proc(n)
    1/(1-u*v-u*w-v*w-u*v*w)/(1-x*y-x*z-y*z) ;
    coeftayl(%, x=0, n) ;
    coeftayl(%, y=0, n) ;
    coeftayl(%, z=0, n) ;
    coeftayl(%, u=0, n) ;
    coeftayl(%, v=0, n) ;
    coeftayl(%, w=0, n) ;
end proc:
seq(A268552(2*n), n=0..40) ; # R. J. Mathar, Mar 10 2016

Mathematica

f = 1/((1 - u v - u w - v w - u v w)*(1 - x y - x z - y z));
a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}]&, f, {x, y, z, u, v, w}];
Table[a[2n], {n, 0, 20}] (* Jean-François Alcover, Dec 03 2017 *)

Crossrefs

Cf. A268545 - A268555.

Sequence in context: A159417 A289540 A275453 * A227493 A095423 A174766

Adjacent sequences: A268549 A268550 A268551 * A268553 A268554 A268555

Keyword

nonn

Author

N. J. A. Sloane, Feb 29 2016

Status

approved