cp's OEIS Frontend

A274783 Diagonal of the rational function 1/(1 - (w*x*y*z + w*x*y + w*x*z + w*y*z + x*y*z)).

%I A274783 #30 Jul 04 2025 10:04:31
%S A274783 1,1,1,25,121,361,3361,24361,116425,790441,6060121,36888721,238815721,
%T A274783 1760983225,11968188961,79763351305,570661612585,4040282139625,
%U A274783 27901708614985,198090585115105,1420583920034161,10056659775872161,71730482491962361,517012699162717825,3713833648541268121
%N A274783 Diagonal of the rational function 1/(1 - (w*x*y*z + w*x*y + w*x*z + w*y*z + x*y*z)).
%C A274783 Diagonal of the rational function 1/(1 - (x^3 + y^3 + z^3 + w^3 + x*y*z*w)). - _Seiichi Manyama_, Jul 04 2025
%H A274783 Seiichi Manyama, <a href="/A274783/b274783.txt">Table of n, a(n) for n = 0..1000</a>
%H A274783 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 A274783 S. 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 A274783 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 A274783 0 = x^2*(x+3)^2*(x^4 - 260*x^3 + 6*x^2 - 4*x + 1)*y''' + 3*x*(x+3)*(2*x^5 - 381*x^4 - 1944*x^3 + 34*x^2 - 18*x + 3)*y'' + (7*x^6 - 764*x^5 - 9101*x^4 - 27264*x^3 + 381*x^2 - 132*x + 9)*y' + (x^5 - 13*x^4 - 246*x^3 - 5946*x^2 + 69*x - 9)*y, where y is the g.f.
%F A274783 a(n) = Sum_{k = 0..floor(n/3)} (n+k)!/(k!^4*(n-3*k)!) = Sum_{k = 0..floor(n/3)} binomial(n,3*k)*binomial(n+k,k)*(3*k)!/k!^3 (apply Eger, Theorem 3 to the set of column vectors S = {[1,1,1,1], [1,1,0,1], [1,0,1,1], [0,1,1,1], [1,1,1,0]}). - _Peter Bala_, Jan 27 2018
%F A274783 G.f.: Sum_{k>=0} (4*k)!/k!^4 * x^(3*k)/(1-x)^(4*k+1). - _Seiichi Manyama_, Mar 19 2023
%F A274783 From _Vaclav Kotesovec_, Mar 19 2023: (Start)
%F A274783 Recurrence: n^3*(2*n - 5)*(4*n - 11)*(4*n - 7)*a(n) = (4*n - 11)*(32*n^5 - 184*n^4 + 368*n^3 - 327*n^2 + 147*n - 27)*a(n-1) - (192*n^6 - 1920*n^5 + 7628*n^4 - 15366*n^3 + 16567*n^2 - 9117*n + 2025)*a(n-2) + (4*n - 9)*(4*n - 3)*(520*n^4 - 4420*n^3 + 13809*n^2 - 18769*n + 9367)*a(n-3) - (n-3)^3*(2*n - 3)*(4*n - 7)*(4*n - 3)*a(n-4).
%F A274783 a(n) ~ sqrt(9/8 + 3/(32*sqrt(2)) + sqrt(1085/32 + 161/(2*sqrt(2)))/8) * (1 + 2*sqrt(2) + 2*sqrt(2*(2*sqrt(2) - 1)))^n / (Pi^(3/2) * n^(3/2)). (End)
%p A274783 with(combinat):
%p A274783 seq(add((n+k)!/(k!^4*(n-3*k)!), k = 0..floor(n/3)), n = 0..20); # _Peter Bala_, Jan 27 2018
%o A274783 (PARI)
%o A274783 my(x='x, y='y, z='z, w='w);
%o A274783 R = 1/(1-(w*x*y*z+w*x*y+w*x*z+w*y*z+x*y*z));
%o A274783 diag(n, expr, var) = {
%o A274783   my(a = vector(n));
%o A274783   for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
%o A274783   for (k = 1, n, a[k] = expr;
%o A274783        for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
%o A274783   return(a);
%o A274783 };
%o A274783 diag(20, R, [x,y,z,w])
%Y A274783 Cf. A268545-A268555.
%Y A274783 Cf. A001850, A208425, A361636.
%Y A274783 Cf. A082488, A274785, A361637.
%K A274783 nonn,easy
%O A274783 0,4
%A A274783 _Gheorghe Coserea_, Jul 13 2016