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 A268555 Diagonal of the rational function of six variables 1/((1 - w - u v - u v w) * (1 - z - x y)). 73
 1, 6, 78, 1260, 22470, 424116, 8305836, 166929048, 3419932230, 71109813060, 1496053026468, 31777397077608, 680354749147164, 14664155597771400, 317877850826299800, 6924815555276838960, 151505459922479997510, 3327336781596164286180 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Also diagonal of rational function R(x,y,z) = 1 /(1 - x - y - z - x*y + x*z). Annihilating differential operator: x*(16*x^2-24*x+1)*(d/dx)^2 + (48*x^2-48*x+1)*(d/dx) + 12*x-6. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..200 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 Conjecture D-finite: n^2*a(n) -6*(2*n-1)^2*a(n-1) +4*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Mar 10 2016 a(n) ~ sqrt(4+3*sqrt(2)) * 2^(2*n-3/2) * (1+sqrt(2))^(2*n) / (Pi*n). - Vaclav Kotesovec, Jul 01 2016 G.f.: hypergeom([1/12, 5/12],[1],6912*x^4*(1-24*x+16*x^2)/(1-24*x+48*x^2)^3)/(1-24*x+48*x^2)^(1/4). 0 = x*(16*x^2-24*x+1)*y'' + (48*x^2-48*x+1)*y' + (12*x-6)*y, where y is g.f. a(n) = A000984(n)*A001850(n) = C(2*n,n)*Sum_{k = 0..n} C(n,k)*C(n+k,k). - Peter Bala, Mar 19 2018 EXAMPLE G.f. = 1 + 6*x + 78*x^2 + 1260*x^3 + 22470*x^4 + 424116*x^5 + 8305836*x^6 + ... MAPLE A268555 := proc(n)     1/(1-w-u*v-u*v*w)/(1-z-x*y) ;     coeftayl(%, x=0, n) ;     coeftayl(%, y=0, n) ;     coeftayl(%, z=0, n) ;     coeftayl(%, u=0, n) ;     coeftayl(%, v=0, n) ;     coeftayl(%, w=0, n) ; end proc: seq(A268555(n), n=0..40) ; # R. J. Mathar, Mar 10 2016 seq(binomial(2*n, n)*add(binomial(n, k)*binomial(n+k, k), k = 0..n), n = 0..20); # Peter Bala, Mar 19 2018 MATHEMATICA sc = SeriesCoefficient; a[n_] := 1/(1-w-u*v-u*v*w)/(1-z-x*y) // sc[#, {x, 0, n}]& // sc[#, {y, 0, n}]& // sc[#, {z, 0, n}]& // sc[#, {u, 0, n}]& // sc[#, {v, 0, n}]& // sc[#, {w, 0, n}]&; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 14 2017 *) a[n_] := Product[Hypergeometric2F1[-n, -n, 1, i], {i, 1, 2}]; Table[a[n], {n, 0, 17}]  (* Peter Luschny, Mar 19 2018 *) PROG (PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi"); read("hypergeom.gpi"); N = 18; x = 'x + O('x^N); Vec(hypergeom([1/12, 5/12], [1], 6912*x^4*(1-24*x+16*x^2)/(1-24*x+48*x^2)^3, N)/(1-24*x+48*x^2)^(1/4)) \\ Gheorghe Coserea, Jul 05 2016 (PARI) {a(n) = if( n<1, n==0, my(A = vector(n+1)); A[1] = 1; A[2] = 6; for(k=2, n, A[k+1] = (6*(2*k-1)^2*A[k] - 4*(2*k-1)*(2*k-3)*A[k-1]) / k^2); A[n+1])}; /* Michael Somos, Jan 22 2017 */ (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), 18) \\ test: diag(1/(1-x-y-z-x*y+x*z)) == diag(1/((1-w-u*v-u*v*w)*(1-z-x*y))) \\ Gheorghe Coserea, Jun 15 2018 (GAP) List([0..20], n->Binomial(2*n, n)*Sum([0..n], k->Binomial(n, k)*Binomial(n+k, k))); # Muniru A Asiru, Mar 19 2018 CROSSREFS Cf. A268545-A268555. Cf. A000984, A001850. Sequence in context: A074112 A197103 A208473 * A131926 A132866 A279444 Adjacent sequences:  A268552 A268553 A268554 * A268556 A268557 A268558 KEYWORD nonn,easy,changed AUTHOR N. J. A. Sloane, Feb 29 2016 STATUS approved

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Last modified January 28 00:32 EST 2020. Contains 331313 sequences. (Running on oeis4.)