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%I #35 Apr 04 2023 11:44:36
%S 1,6,114,2940,87570,2835756,96982116,3446781624,126047377170,
%T 4712189770860,179275447715364,6918537571788024,270178056420497316,
%U 10656693484898995800,423937118582497715400,16989669600664370275440,685277433339552643145490,27797911234749454227812460,1133299570662800455270517700
%N Diagonal of the rational function 1/(1-(wxz + wy + wz + xy + xz + y + z)).
%H Gheorghe Coserea, <a href="/A274786/b274786.txt">Table of n, a(n) for n = 0..200</a>
%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 Timothy Huber, Daniel Schultz, and Dongxi Ye, <a href="https://doi.org/10.4064/aa220621-19-12">Ramanujan-Sato series for 1/pi</a>, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
%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 a(n) = Sum_{j=0..2n} (-1)^j * binomial(2*n,j) * binomial(j,n)^3.
%F a(n) = T(2*n,n), where triangle T(n,k) is defined by A262704.
%F 0 = (-x^2+44*x^3+16*x^4)*y''' + (-3*x+198*x^2+96*x^3)*y'' + (-1+144*x+108*x^2)*y' + (6+12*x)*y, where y is the g.f.
%F Recurrence: n^3*a(n) = 2*(2*n - 1)*(11*n^2 - 11*n + 3)*a(n-1) + 4*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2). - _Vaclav Kotesovec_, Dec 01 2017
%F a(n) ~ 2^(2*n - 1) * phi^(5*n + 5/2) / (5^(1/4) * (Pi*n)^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, Dec 01 2017
%F Conjecture: a(n) = [x^n] (1 + x)^(2*n) * P(n,(1 + x)/(1 - x))^2, where P(n,x) denotes the n-th Legendre polynomial. Cf. A005260(n) = [x^n] (1 - x)^(2*n) * P(n,(1 + x)/(1 - x))^2, due to Carlitz. - _Peter Bala_, Sep 21 2021
%t a[n_] := Sum[(-1)^j Binomial[2n, j] Binomial[j, n]^3, {j, n, 2n}];
%t (* or much faster *)
%t a[0] = 1; a[1] = 6; a[n_] := a[n] = (2*(2*n - 1)*(11*n^2 - 11*n + 3)*a[n - 1] + 4*(n - 1)*(2*n - 3)*(2*n - 1)*a[n - 2])/n^3;
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Dec 01 2017, after _Vaclav Kotesovec_ *)
%o (PARI)
%o a(n) = sum(j=n, 2*n, (-1)^(j)*binomial(2*n, 2*n - j)*binomial(j, n)^3);
%o (PARI)
%o my(x='x, y='y, z='z, w='w);
%o R = 1/(1-(w*x*z+w*y+w*z+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(18, R, [x,y,z,w])
%Y Cf. A262704, A268545-A268555.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Jul 14 2016
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