A274784 - OEIS

A274784

Diagonal of the rational function 1/(1-(wxyz + wxy + wxz + wyz + xz + yz)).

1, 1, 13, 49, 661, 4021, 49141, 382033, 4387909, 39174421, 435947953, 4223039173, 46369242349, 472149790477, 5166034019173, 54291944826769, 594930602047909, 6384197552441317, 70225109775221041, 764509591210501189, 8448837429195472441, 92938878099911462521, 1032029676943832312473

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OFFSET

0,3

LINKS

Table of n, a(n) for n=0..22.

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.

Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

FORMULA

0 = x^2*(1-x)*(x+2)^2*(x^3+105*x^2+3*x-1)*y''' - 6*x*(x+2)*(x^5+81*x^4+209*x^3-205*x^2-6*x+1)*y'' - (7*x^6+378*x^5+2463*x^4+3904*x^3-2808*x^2-60*x+4)*y' + (-x^5-19*x^4-112*x^3-1196*x^2+352*x+4)*y, where y is the g.f.

a(n) = Sum_{i = 0..floor(n/2)} Sum_{j = 0..floor(n/2)-i} (n+i+j)!/((n-2*i-2*j)!*(i+j)!*i!^2*j!^2). - Peter Bala, Mar 16 2023

From Vaclav Kotesovec, Mar 17 2023: (Start)

Recurrence: n^3*(3*n - 8)*(3*n - 7)*(3*n - 5)*a(n) = 2*(3*n - 8)*(18*n^5 - 99*n^4 + 190*n^3 - 162*n^2 + 69*n - 12)*a(n-1) + 3*(3*n - 4)*(306*n^5 - 2550*n^4 + 8065*n^3 - 11991*n^2 + 8306*n - 2132)*a(n-2) - (3*n - 7)*(3*n - 2)*(312*n^4 - 2548*n^3 + 7614*n^2 - 9843*n + 4644)*a(n-3) - (n-3)^3*(3*n - 5)*(3*n - 4)*(3*n - 2)*a(n-4).

a(n) ~ c * (1 + 12*cos(arccos(1/4)/3))^n / (Pi^(3/2) * n^(3/2)), where c = sqrt(49/48 + sqrt(199)*cos(arccos(-163/(796*sqrt(199)))/3)/12). (End)

MAPLE

seq(add(add( (n+i+j)!/((n-2*i-2*j)!*(i+j)!*i!^2*j!^2), j = 0..floor(n/2)-i), i = 0..floor(n/2)), n = 0..22); # Peter Bala, Mar 16 2023

PROG

(PARI)

my(x='x, y='y, z='z, w='w);

R = 1/(1-(w*x*y*z+w*x*y+w*x*z+w*y*z+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)));