A268543 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

From Gheorghe Coserea, Jul 03 2016: (Start)

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

Annihilating differential operator: x*(2*x+3)*(16*x^2-71*x+2)*Dx^2 + 2*(32*x^3+x^2-213*x+3)*Dx + 8*x^2+48*x-48.

(End)

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"

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.

a(n) = A000984(n)*A114496(n). (End)

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

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 *)

PROG

(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])