A268543 The diagonal of 1/(1 - (y + z + x z + x w + x y w)).
1, 8, 156, 3800, 102340, 2919168, 86427264, 2626557648, 81380484900, 2559296511200, 81443222791216, 2616761264496288, 84749038859067856, 2763262653898544000, 90615128199047200800, 2986287891921565639200, 98841887070519004625700
Offset: 0
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.
- S. 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
A268543 := proc(n) 1/(1-y-z-x*z-x*w-x*y*w) ; coeftayl(%,x=0,n) ; coeftayl(%,y=0,n) ; coeftayl(%,z=0,n) ; coeftayl(%,w=0,n) ; end proc: seq(A268543(n),n=0..40) ; # R. J. Mathar, Mar 11 2016 #alternative program with(combinat): seq(binomial(2*n,n)*add(binomial(n,k)*binomial(2*n+k,k), k = 0..n), n = 0..20); # Peter Bala, Jan 27 2018
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Mathematica
CoefficientList[Series[HypergeometricPFQ[{1/12, 5/12}, {1}, 1728*x^3*(2 - 71*x + 16*x^2)/(1 - 32*x + 16*x^2)^3]*(1 - 32*x + 16*x^2)^(-1/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 05 2016 *) -
PARI
my(x='x, y='y, z='z, w='w); R = 1/(1 - x - y - z - x*y); 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 = 20; x = 'x + O('x^N); Vec(hypergeom_sym([1/12,5/12],[1],1728*x^3*(16*x^2-71*x+2)/(16*x^2-32*x+1)^3, N)/(16*x^2-32*x+1)^(1/4)) \\ Gheorghe Coserea, Jul 03 2016
Formula
Conjecture: 2*n^2*(17*n-23)*a(n) +(-1207*n^3+2840*n^2-1897*n+360)*a(n-1) + 4*(17*n-6)*(-3+2*n)^2*a(n-2) = 0. - R. J. Mathar, Mar 11 2016
G.f.: hypergeom([1/12, 5/12], [1], 1728*x^3*(2-71*x+16*x^2)/(1-32*x+16*x^2)^3)*(1-32*x+16*x^2)^(-1/4). - Gheorghe Coserea, Jul 01 2016
0 = x*(2*x+3)*(16*x^2-71*x+2)*y'' + 2*(32*x^3+x^2-213*x+3)*y' + (8*x^2+48*x-48)*y, where y is the g.f. - Gheorghe Coserea, Jul 03 2016
a(n) ~ sqrt(3 + 13/sqrt(17)) * (71+17*sqrt(17))^n / (Pi * n * 2^(2*n + 3/2)). - Vaclav Kotesovec, Jul 05 2016
From Peter Bala, Jan 27 2018: (Start)
a(n) = binomial(2*n,n)*Sum_{k = 0..n} binomial(n,k)* binomial(2*n+k,k) (apply Eger, Theorem 3 to the set of column vectors S = {[1,0,0], [0,1,0], [0,0,1], [1,1,0]}). Using this binomial sum, Maple confirms the above recurrence of Mathar.
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