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 A274671 Diagonal of the rational function 1/(1 - x - y - z - x y + x z + y z - x y z). 1
 1, 5, 49, 593, 7921, 111965, 1641865, 24705665, 378931585, 5898329045, 92893693729, 1477015762865, 23671209428881, 381902943661517, 6196712221450009, 101043805539177473, 1654726991239056385, 27201210101330189477, 448652354194417534609, 7422254208909904273553 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Annihilating differential operator: x*(4*x-5)*(x^3+27*x^2-19*x+1)*Dx^2 + (12*x^4+196*x^3-481*x^2+190*x-5)*Dx + 4*x^3+18*x^2-95*x+25. Also diagonal of rational functions 1/(1 + z + 2*x*y - y*z + x*z - x*y*z), 1/(1 + 2*z + x*y - 2*y*z + x*z - x*y*z), 1/(1 + y + z + 2*x*y + y*z + x*z + x*y*z), 1/(1 + y + 2*z + x*y + 2*y*z + x*z + x*y*z), 1/(1 + x + y - 2*z - y*z - 2*x*z + x*y*z). - Gheorghe Coserea, Jul 03 2018 LINKS Gheorghe Coserea, Table of n, a(n) for n = 0..310 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 G.f.: hypergeom([1/12, 5/12],[1],13824*x^5*(1-19*x+27*x^2+x^3)/(1-20*x+54*x^2+28*x^3+x^4)^3)/(1-20*x+54*x^2+28*x^3+x^4)^(1/4). 0 = x*(4*x-5)*(x^3+27*x^2-19*x+1)*y'' + (12*x^4+196*x^3-481*x^2+190*x-5)*y' + (4*x^3+18*x^2-95*x+25)*y, where y is the g.f. MATHEMATICA gf = Hypergeometric2F1[1/12, 5/12, 1, 13824*x^5*(1 - 19*x + 27*x^2 + x^3) / (1 - 20*x + 54*x^2 + 28*x^3 + x^4)^3]/(1 - 20*x + 54*x^2 + 28*x^3 + x^4)^(1/4); CoefficientList[gf + O[x]^20, x] (* Jean-François Alcover, Dec 01 2017 *) PROG (PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi"); read("hypergeom.gpi"); N = 20; x = 'x + O('x^N); Vec(hypergeom([1/12, 5/12], [1], 13824*x^5*(1-19*x+27*x^2+x^3)/(1-20*x+54*x^2+28*x^3+x^4)^3, N)/(1-20*x+54*x^2+28*x^3+x^4)^(1/4)) (PARI) diag(expr, N=22, var=variables(expr)) = {   my(a = vector(N));   for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));   for (n = 1, N, a[n] = expr;     for (k = 1, #var, a[n] = polcoeff(a[n], n-1)));   return(a); }; diag(1/(1 - x - y - z - x*y + x*z + y*z - x*y*z), 20) \\ test: diag(1/(1 - x - y - z - x*y + x*z + y*z - x*y*z)) == diag(1/(1 + x + y - 2*z - y*z - 2*x*z + x*y*z)) \\ Gheorghe Coserea, Jul 03 2018 CROSSREFS Cf. A268545-A268555. Sequence in context: A212818 A195206 A081474 * A112241 A216483 A243945 Adjacent sequences:  A274668 A274669 A274670 * A274672 A274673 A274674 KEYWORD nonn AUTHOR Gheorghe Coserea, Jul 05 2016 STATUS approved

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Last modified November 25 11:32 EST 2020. Contains 338623 sequences. (Running on oeis4.)