|
%I #39 Oct 27 2023 21:46:52
%S 1,18,1350,141120,17151750,2272538268,318430816704,46404203788800,
%T 6961609406993670,1068002895589987500,166779781860762170100,
%U 26422986893371642828800,4236593267629481817240000,686167053247777413372681600,112093956900827388909570240000
%N Diagonal of the rational function 1/(1-(1+w)(xy + xz + yz)) [even-indexed terms only].
%C Odd-order terms are zero since R(x,y,z,w) = R(-x,-y,-z,w), where R(x,y,z,w) = 1/(1-(1+w)*(x*y + x*z + y*z)).
%C From _Peter Bala_, Jun 22 2023: (Start)
%C a(n) = A(n,n,2*n,2*n) (= A(2*n,2*n,n,n)) in the notation of Straub, equation 8, where it is shown that 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. This also follows from Meštrović equation 39, since a(n) = binomial(3*n,n)^2 * binomial(2*n,n).
%C Inductively define a family of sequences {a(i,n) : n >= 0}, i >= 1, by setting a(1,n) = a(n) and, for i >= 2, a(i,n) = [x^n] ( exp(Sum_{k >= 1} a(i-1,k)*x^k/k) )^n.
%C We conjecture that the sequences {a(i,n) : n >= 0}, i >= 2, also satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5, and positive integers n and r. Cf. A362725 and A362732. (End)
%H Alois P. Heinz, <a href="/A275047/b275047.txt">Table of n, a(n) for n = 0..444</a> (first 34 terms from Gheorghe Coserea)
%H 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.
%H R. Meštrović, <a href="https://arxiv.org/abs/1111.3057">Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012)</a>, arXiv:1111.3057 [math.NT], 2011.
%H Armin Straub, <a href="http://dx.doi.org/10.2140/ant.2014.8.1985">Multivariate Apéry numbers and supercongruences of rational functions</a>, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; <a href="https://arxiv.org/abs/1401.0854">arXiv preprint</a>, arXiv:1401.0854 [math.NT], 2014.
%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 0 = (-4*x^2+729*x^4)*y'''' + (-20*x+7290*x^3)*y''' + (-16+18063*x^2)*y'' + 10449*x*y' + 576*y, where y = 1 + 18*x^2 + 1350*x^4 + ...
%F From _Vaclav Kotesovec_, Aug 03 2016: (Start)
%F a(n) = (3*n)!^2 / (n!^4 * (2*n)!).
%F a(n) ~ 3^(6*n+1) / (Pi^(3/2) * n^(3/2) * 2^(2*n+2)).
%F (End)
%F G.f.: 4F3(1/3,1/3,2/3,2/3;1/2,1,1;729x/4). - _Benedict W. J. Irwin_, Aug 05 2016
%F From _Peter Bala_, Sep 20 2021: (Start)
%F a(n) = 9*(3*n - 1)^2*(3*n - 2)^2/(2*n^3*(2*n - 1))*a(n-1).
%F a(n) = Sum_{k = n..3*n} (-1)^k*binomial(3*n,k)^2*binomial(k,n)^2. (End)
%F From _Peter Bala_, Jun 22 2023: (Start)
%F a(n) = binomial(3*n,n)^2 * binomial(2*n,n) = A188662(n) * A000984(n).
%F a(n) = Sum_{k = 0..n} binomial(n,k)*binomial(2*n,k)*binomial(2*n-k,n)* binomial(4*n-k,2*n).
%F a(n) = [(x*y)^n * (z*t)^(2*n)] 1/((1 - x - y)*(1 - z - t) - x*y*z*t). (End)
%e 1 + 18*x^2 + 1350*x^4 + 141120*x^6 + ...
%p a:= proc(n) option remember; `if`(n=0, 1,
%p 9*(3*n-1)^2*(3*n-2)^2*a(n-1)/((4*n-2)*n^3))
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Jul 25 2016
%t Table[(3*n)!^2 / (n!^4*(2*n)!), {n, 0, 20}] (* _Vaclav Kotesovec_, Aug 03 2016 *)
%t CoefficientList[Series[HypergeometricPFQ[{1/3, 1/3, 2/3, 2/3}, {1/2, 1, 1}, 729x/4], {x, 0, 10}], x] (* _Benedict W. J. Irwin_, Aug 05 2016 *)
%o (PARI)
%o my(x='x, y='y, z='z, w='w);
%o R = 1/(1-(1+w)*(x*y+x*z+y*z));
%o diag(n, expr, var) = {
%o my(a = vector(n));
%o for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
%o for (k = 1, n, a[k] = expr;
%o for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
%o return(a);
%o };
%o diag(23, R, [x,y,z,w])
%Y Cf. A000984, A188662, A268545 - A268555.
%K nonn,easy
%O 0,2
%A _Gheorghe Coserea_, Jul 18 2016
|
