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 A275047 Diagonal of the rational function 1/(1-(1+w)(xy + xz + yz)) [even-indexed terms only]. 2
 1, 18, 1350, 141120, 17151750, 2272538268, 318430816704, 46404203788800, 6961609406993670, 1068002895589987500, 166779781860762170100, 26422986893371642828800, 4236593267629481817240000, 686167053247777413372681600, 112093956900827388909570240000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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)). LINKS Gheorghe Coserea and Alois P. Heinz, Table of n, a(n) for n = 0..444 (first 34 terms from Gheorghe Coserea) A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015. FORMULA 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 + ... From Vaclav Kotesovec, Aug 03 2016: (Start) a(n) = (3*n)!^2 / (n!^4 * (2*n)!). a(n) ~ 3^(6*n+1) / (Pi^(3/2) * n^(3/2) * 2^(2*n+2)). (End) G.f.: 4F3(1/3,1/3,2/3,2/3;1/2,1,1;729x/4). - Benedict W. J. Irwin, Aug 05 2016 EXAMPLE 1 + 18*x^2 + 1350*x^4 + 141120*x^6 + ... MAPLE a:= proc(n) option remember; `if`(n=0, 1,       9*(3*n-1)^2*(3*n-2)^2*a(n-1)/((4*n-2)*n^3))     end: seq(a(n), n=0..20);  # Alois P. Heinz, Jul 25 2016 MATHEMATICA Table[(3*n)!^2 / (n!^4*(2*n)!), {n, 0, 20}] (* Vaclav Kotesovec, Aug 03 2016 *) 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 *) PROG my(x='x, y='y, z='z, w='w); R = 1/(1-(1+w)*(x*y+x*z+y*z)); diag(n, expr, var) = {   my(a = vector(n));   for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));   for (k = 1, n, a[k] = expr;        for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));   return(a); }; diag(23, R, [x, y, z, w]) CROSSREFS Cf. A268545-A268555. Sequence in context: A252969 A182286 A292609 * A160252 A276015 A210823 Adjacent sequences:  A275044 A275045 A275046 * A275048 A275049 A275050 KEYWORD nonn AUTHOR Gheorghe Coserea, Jul 18 2016 STATUS approved

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Last modified July 31 22:27 EDT 2021. Contains 346377 sequences. (Running on oeis4.)