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 A274786 Diagonal of the rational function 1/(1-(wxz + wy + wz + xy + xz + y + z)). 2
 1, 6, 114, 2940, 87570, 2835756, 96982116, 3446781624, 126047377170, 4712189770860, 179275447715364, 6918537571788024, 270178056420497316, 10656693484898995800, 423937118582497715400, 16989669600664370275440, 685277433339552643145490, 27797911234749454227812460, 1133299570662800455270517700 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Gheorghe Coserea, 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 a(n) = Sum_{j=0..2n} (-1)^j * binomial(2*n,j) * binomial(j,n)^3. a(n) = T(2*n,n), where triangle T(n,k) is defined by A262704. 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. 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 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 MATHEMATICA a[n_] := Sum[(-1)^j Binomial[2n, j] Binomial[j, n]^3, {j, n, 2n}]; (* or (much faster): *) 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; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 01 2017, after Vaclav Kotesovec *) PROG (PARI) a(n) = sum(j=n, 2*n, (-1)^(j)*binomial(2*n, 2*n - j)*binomial(j, n)^3); (PARI) my(x='x, y='y, z='z, w='w); R = 1/(1-(w*x*z+w*y+w*z+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(18, R, [x, y, z, w]) CROSSREFS Cf. A262704, A268545-A268555. Sequence in context: A194476 A059116 A121544 * A317172 A278752 A003425 Adjacent sequences:  A274783 A274784 A274785 * A274787 A274788 A274789 KEYWORD nonn AUTHOR Gheorghe Coserea, Jul 14 2016 STATUS approved

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Last modified January 19 21:47 EST 2020. Contains 331066 sequences. (Running on oeis4.)