A268545 From the diagonal of 1/(1 - (y + z + x w + x z w + x y w)).
1, 10, 246, 7540, 255430, 9163980, 341237820, 13042646760, 508236930630, 20101587623260, 804500381097556, 32508382071448920, 1324112273705453596, 54296281503438398200, 2239266766596344681400, 92809720054802928741840, 3863305447624183692730950, 161427619265399264526790140
Offset: 0
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.
- 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
A268545 := proc(n) 1/(1-y-z-x*w-x*z*w-x*y*w) ; coeftayl(%, x=0, n) ; coeftayl(%, y=0, n) ; coeftayl(%, z=0, n) ; coeftayl(%, w=0, n) ; end proc: seq(A268545(n), n=0..40) ; # R. J. Mathar, Apr 15 2016 #alternative program with(combinat): seq(add(binomial(n,j)*add(binomial(j,k)*binomial(n+k,j)*binomial(2*n+k,n), k = 0..j), j = 0..n), n = 0..20); # Peter Bala, Jan 26 2018
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Mathematica
a[n_] := a[n] = 1/(1 - y - z - x*w - x*z*w - x*y*w) // SeriesCoefficient[#, {x, 0, n}]& // SeriesCoefficient[#, {y, 0, n}]& // SeriesCoefficient[#, {z, 0, n}]& // SeriesCoefficient[#, {w, 0, n}]&; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 40}] (* or: *) HypergeometricPFQ[{1/12, 5/12}, {1}, (6912*x^3*(-16*x^2 - 44*x + 1))/ (16*x^2 - 40*x + 1)^3]/(16*x^2 - 40*x + 1)^(1/4) + O[x]^41 // CoefficientList[#, x]& (* Jean-François Alcover, Nov 12 2017, after Gheorghe Coserea *) -
PARI
\\ system("wget http://www.jjj.de/pari/hypergeom.gpi"); read("hypergeom.gpi"); N = 21; x = 'x + O('x^N); Vec(hypergeom([1/12, 5/12],[1],6912*x^3*(1-44*x-16*x^2)/(1-40*x+16*x^2)^3, N)/(1-40*x+16*x^2)^(1/4)) \\ Gheorghe Coserea, Jul 03 2016 -
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 - y*z), 18) \\ test: diag(1/(1-x-y-z-x*y-y*z)) == diag(1/(1-(y+z+x*w + x*z*w + x*y*w))) \\ Gheorghe Coserea, Jun 16 2018
Formula
D-finite with recurrence: n^2*(10*n-13)*a(n) +2*(-220*n^3+506*n^2-334*n+63)*a(n-1) -4*(10*n-3)*(-3+2*n)^2*a(n-2)=0. - R. J. Mathar, Apr 15 2016
a(n) ~ (1+sqrt(5))^(5*n+2) / (5^(1/4) * Pi * n * 2^(3*n+3)). - Vaclav Kotesovec, Jul 01 2016
G.f.: hypergeom([1/12, 5/12],[1],6912*x^3*(1-44*x-16*x^2)/(1-40*x+16*x^2)^3)/(1-40*x+16*x^2)^(1/4). - Gheorghe Coserea, Jul 01 2016
0 = x*(4*x+3)*(16*x^2+44*x-1)*y'' + (128*x^3+320*x^2+264*x-3)*y' + (16*x^2+12*x+30)*y, where y is the g.f. - Gheorghe Coserea, Jul 03 2016
a(n) = Sum_{j = 0..n} C(n,j)*Sum_{k = 0..j} C(j,k)*C(n+k,j)*C(2*n+k,n) (apply Eger, Theorem 3 to the set of column vectors S = {[1,0,0], [0,1,0], [0,0,1], [1,1,0], [0,1,1]}. - Peter Bala, Jan 26 2018
Comments