%I #32 Oct 27 2023 18:29:50
%S 1,30,3870,698880,146881350,33664290660,8161119046080,
%T 2057838873742080,534207274296708870,141819633092233024500,
%U 38325111684989079770820,10507683011039968126464000,2915637485437466384104036800,817228372649330361131497411200
%N Diagonal of the rational function 1/(1 - (w*x*z + x*y*z + w*y + x*y + z)) [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 - (w*x*z + x*y*z + w*y + x*y + z)).
%H Alois P. Heinz, <a href="/A275049/b275049.txt">Table of n, a(n) for n = 0..402</a> (first 34 terms from Gheorghe Coserea)
%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 [math-ph], 2015.
%H S. Eger, <a href="http://arxiv.org/abs/1511.00622">On the Number of Many-to-Many Alignments of N Sequences</a>, arXiv:1511.00622 [math.CO], 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 a(n) = [(x*y*z*w)^(2n)] 1/(1 - (w*x*z + x*y*z + w*y + x*y + z)).
%F 0 = (10624*x^3 - 3427384*x^5 + 24111324*x^7 - 10234350*x^9 - 1476225*x^11)*y'''' + (42496*x^2 - 30355056*x^4 + 182389536*x^6 - 140729400*x^8 - 14762250*x^10)*y''' + (10624*x - 62747592*x^3 + 309393468*x^5 - 459003510*x^7 - 36577575*x^9)*y'' + (-10624 - 25151544*x^2 + 117851940*x^4 - 351676890*x^6 - 21159225*x^8)*y' + (11243520*x^3 - 29496960*x^5 - 1166400*x^7)*y, where y(x) = A(x^2).
%F From _Vaclav Kotesovec_, Aug 03 2016: (Start)
%F Recurrence: 4*(n-1)*n^3*(238*n^2 - 663*n + 457)*a(n) = 2*(n-1)*(150416*n^5 - 644640*n^4 + 1020351*n^3 - 734334*n^2 + 237007*n - 26880)*a(n-1) - 9*(3*n - 5)^2*(3*n - 4)^2*(238*n^2 - 187*n + 32)*a(n-2).
%F a(n) ~ sqrt(1 + 11/(4*sqrt(7))) * (158 + 119*sqrt(7)/2)^n / (2*Pi^(3/2)*n^(3/2)).
%F (End)
%F a(n) = Sum_{k = 0..n} C(3*n+k,k)*C(n,k)*(3*n)!/n!^3 (apply Eger, Theorem 3 to the set of column vectors S = {[0,0,1,0], [1,1,0,0], [0,1,0,1], [1,1,1,0], [1,0,1,1]}). - _Peter Bala_, Jan 27 2018
%e 1 + 30*x^2 + 3870*x^4 + 698880*x^6 + ...
%p with(combinat):
%p seq(add(binomial(3n+k,k)*binomial(n,k)*(3n)!/n!^3, k = 0..n), n = 0..20); # _Peter Bala_, Jan 27 2018
%t a[n_] := (3n)! Hypergeometric2F1[-n, 3n + 1, 1, -1]/n!^3;
%t a /@ Range[0, 20] (* _Jean-François Alcover_, Feb 29 2020, after _Peter Bala_ *)
%o (PARI)
%o my(x='x, y='y, z='z, w='w);
%o R = 1/(1-(w*x*z+x*y*z+w*y+x*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. A268545-A268555.
%K nonn,easy
%O 0,2
%A _Gheorghe Coserea_, Jul 19 2016