%I #47 Feb 21 2020 07:24:23

%S 1,10,246,7540,255430,9163980,341237820,13042646760,508236930630,

%T 20101587623260,804500381097556,32508382071448920,1324112273705453596,

%U 54296281503438398200,2239266766596344681400,92809720054802928741840,3863305447624183692730950,161427619265399264526790140

%N From the diagonal of 1/(1 - (y + z + x w + x z w + x y w)).

%C From _Gheorghe Coserea_, Jul 03 2016: (Start)

%C Also diagonal of R(x,y,z) = 1/(1 - x - y - z - x*y - y*z).

%C Annihilating differential operator: x*(4*x+3)*(16*x^2+44*x-1)*Dx^2 + (128*x^3+320*x^2+264*x-3)*Dx + 16*x^2+12*x+30.

%C (End)

%H Vaclav Kotesovec, <a href="/A268545/b268545.txt">Table of n, a(n) for n = 0..200</a>

%H A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015.

%H Steffen Eger, <a href="http://arxiv.org/abs/1511.00622">On the Number of Many-to-Many Alignments of N Sequences</a>, arXiv:1511.00622 [math.CO], 2015.

%H Jacques-Arthur Weil, <a href="http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/">Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"</a>

%F 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

%F a(n) ~ (1+sqrt(5))^(5*n+2) / (5^(1/4) * Pi * n * 2^(3*n+3)). - _Vaclav Kotesovec_, Jul 01 2016

%F 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

%F 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

%F 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

%p A268545 := proc(n)

%p 1/(1-y-z-x*w-x*z*w-x*y*w) ;

%p coeftayl(%, x=0, n) ;

%p coeftayl(%, y=0, n) ;

%p coeftayl(%, z=0, n) ;

%p coeftayl(%, w=0, n) ;

%p end proc:

%p seq(A268545(n), n=0..40) ; # _R. J. Mathar_, Apr 15 2016

%p #alternative program

%p 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

%t 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}]&;

%t Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 40}]

%t (* or: *)

%t 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_ *)

%o (PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");

%o read("hypergeom.gpi");

%o N = 21; x = 'x + O('x^N);

%o 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

%o (PARI)

%o diag(expr, N=22, var=variables(expr)) = {

%o my(a = vector(N));

%o for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));

%o for (n = 1, N, a[n] = expr;

%o for (k = 1, #var, a[n] = polcoeff(a[n], n-1)));

%o return(a);

%o };

%o diag(1/(1 - x - y - z - x*y - y*z), 18)

%o \\ test: diag(1/(1-x-y-z-x*y-y*z)) == diag(1/(1-(y+z+x*w + x*z*w + x*y*w)))

%o \\ _Gheorghe Coserea_, Jun 16 2018

%Y Cf. A268545 - A268555.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Feb 29 2016