A268550 Diagonal of 1/(1 - x - y - z - x y - x z - y z).
1, 12, 366, 13800, 574650, 25335072, 1159174464, 54435558672, 2606102756730, 126634810078920, 6226427808402516, 309095505195676992, 15466884541698962736, 779158533743408851200, 39476348002042199114400, 2010009672816216740255520
Offset: 0
Links
- Gheorghe Coserea, Table of n, a(n) for n = 0..310, First 201 terms from Vaclav Kotesovec.
- A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
- Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.
- Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"
Programs
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Maple
A268550 := proc(n) 1/(1-x-y-z-x*y-x*z-y*z) ; coeftayl(%,x=0,n) ; coeftayl(%,y=0,n) ; coeftayl(%,z=0,n) ; end proc: seq(A268550(n),n=0..20) ; # R. J. Mathar, Mar 10 2016
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Mathematica
gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*x^3*(x + 2)^3*(27*x^2 + 54*x - 1)/(-1 + 48*x + 24*x^2)^3]/(1 - 48*x - 24*x^2)^(1/4); CoefficientList[gf + O[x]^20, x] (* Jean-François Alcover, Dec 03 2017, after Gheorghe Coserea *)
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PARI
my(x='x, y='y, z='z); R = 1/(1 - x - y - 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(10, R, [x,y,z]) -
PARI
\\ system("wget http://www.jjj.de/pari/hypergeom.gpi"); read("hypergeom.gpi"); N = 16; x = 'x + O('x^N); Vec(hypergeom([1/12, 5/12], [1], 1728*x^3*(x+2)^3*(27*x^2+54*x-1)/(-1+48*x+24*x^2)^3, N)/(1-48*x-24*x^2)^(1/4)) \\ Gheorghe Coserea, Jul 03 2016
Formula
Conjecture: 2*n^2*(n-1)*a(n) -(n-1)*(107*n^2-107*n+24)*a(n-1) +12*(-9*n^3+27*n^2-22*n+2)*a(n-2) -3*n*(3*n-5)*(3*n-7)*a(n-3)=0. - R. J. Mathar, Mar 10 2016
a(n) ~ sqrt(3) * (27 + 6*sqrt(21))^n / (2*Pi*n). - Vaclav Kotesovec, Jul 01 2016
G.f.: hypergeom([1/12, 5/12], [1], 1728*x^3*(x+2)^3*(27*x^2+54*x-1)/(-1+48*x+24*x^2)^3)/(1-48*x-24*x^2)^(1/4). - Gheorghe Coserea, Jul 01 2016
0 = x*(x+2)*(x+1)*(27*x^2+54*x-1)*y'' + (81*x^4+324*x^3+431*x^2+214*x-2)*y' + (24*x^3+72*x^2+72*x+24)*y, where y is g.f. - Gheorghe Coserea, Jul 03 2016
a(n) = Sum_{i = 0..n} Sum_{k = 0..n-i} Sum_{j = 0..n-k} (n+i+j+k)!/(i!*j!*k!*(n-k-i)!*(n-k-j)!*(2*k+i+j-n)!). Apply [Eger, Theorem 3] to the set of column vectors S = {[1,0,0], [0,1,0], [0,0,1], [1,1,0], [1,0,1], [0,1,1]}. - Peter Bala, Apr 28 2022
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