A274787 - OEIS

%I #13 Mar 19 2023 11:14:05

%S 1,7,109,2347,59161,1630447,47569789,1443991003,45131425129,

%T 1442569728607,46937386025509,1549447085182003,51764874058957801,

%U 1746923396870038303,59463803615708931709,2039212029425398630747,70386791601625688812009,2443437949352620443258463,85253533514289277890811669

%N Diagonal of the rational function 1/(1-(wxyz + wxy + wxz + wy + xz + y + z)).

%H Vaclav Kotesovec, <a href="/A274787/b274787.txt">Table of n, a(n) for n = 0..630</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 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 0 = (-x^2+42*x^3-143*x^4-372*x^5-143*x^6+42*x^7-x^8)*y''' + (-3*x+195*x^2-1116*x^3-1674*x^4-171*x^5+183*x^6-6*x^7)*y'' +  (-1+148*x-1565*x^2-560*x^3+325*x^4+124*x^5-7*x^6)*y' + (7-213*x+278*x^2+22*x^3+3*x^4-x^5)*y, where y is g.f.

%F From _Vaclav Kotesovec_, Mar 19 2023: (Start)

%F Recurrence: (n-2)*n^3*(2*n - 5)*a(n) = (2*n - 5)*(2*n - 1)*(22*n^3 - 66*n^2 + 50*n - 13)*a(n-1) - (2*n - 3)*(230*n^4 - 1380*n^3 + 2790*n^2 - 2160*n + 541)*a(n-2) + (2*n - 5)*(2*n - 1)*(22*n^3 - 132*n^2 + 248*n - 137)*a(n-3) - (n-3)^3*(n-1)*(2*n - 1)*a(n-4).

%F a(n) ~ 10^(1/4) * (19 + 6*sqrt(10))^(n + 1/2) / (8 * Pi^(3/2) * n^(3/2)). (End)

%t a[n_] := SeriesCoefficient[1/(1 - (w x y z + w x y + w x z + w y + x z + y + z)), {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}];

%t Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Nov 16 2018 *)

%o (PARI)

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

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

%o diag(n, expr, var) = {

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

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

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

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

%o   return(a);

%o };

%o diag(12, R, [x,y,z,w])

%Y Cf. A268545-A268555.

%K nonn

%O 0,2

%A _Gheorghe Coserea_, Jul 14 2016