A275049 Diagonal of the rational function 1/(1 - (w*x*z + x*y*z + w*y + x*y + z)) [even-indexed terms only].

1, 30, 3870, 698880, 146881350, 33664290660, 8161119046080, 2057838873742080, 534207274296708870, 141819633092233024500, 38325111684989079770820, 10507683011039968126464000, 2915637485437466384104036800, 817228372649330361131497411200

Offset

0,2

Comments

Odd-order terms are zero since R(x,y,z,w) = R(-x,-y,z,-w), where R(x,y,z,w) = 1/(1 - (w*x*z + x*y*z + w*y + x*y + z)).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..402 (first 34 terms from Gheorghe Coserea)

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

a(n) = [(x*y*z*w)^(2n)] 1/(1 - (w*x*z + x*y*z + w*y + x*y + z)).

0 = (10624*x^3 - 3427384*x^5 + 24111324*x^7 - 10234350*x^9 - 1476225*x^11)*y'''' + (42496*x^2 - 30355056*x^4 + 182389536*x^6 - 140729400*x^8 - 14762250*x^10)*y''' + (10624*x - 62747592*x^3 + 309393468*x^5 - 459003510*x^7 - 36577575*x^9)*y'' + (-10624 - 25151544*x^2 + 117851940*x^4 - 351676890*x^6 - 21159225*x^8)*y' + (11243520*x^3 - 29496960*x^5 - 1166400*x^7)*y, where y(x) = A(x^2).

From Vaclav Kotesovec, Aug 03 2016: (Start)

Recurrence: 4*(n-1)*n^3*(238*n^2 - 663*n + 457)*a(n) = 2*(n-1)*(150416*n^5 - 644640*n^4 + 1020351*n^3 - 734334*n^2 + 237007*n - 26880)*a(n-1) - 9*(3*n - 5)^2*(3*n - 4)^2*(238*n^2 - 187*n + 32)*a(n-2).

a(n) ~ sqrt(1 + 11/(4*sqrt(7))) * (158 + 119*sqrt(7)/2)^n / (2*Pi^(3/2)*n^(3/2)).

(End)

a(n) = Sum_{k = 0..n} C(3*n+k,k)*C(n,k)*(3*n)!/n!^3 (apply Eger, Theorem 3 to the set of column vectors S = {[0,0,1,0], [1,1,0,0], [0,1,0,1], [1,1,1,0], [1,0,1,1]}). - Peter Bala, Jan 27 2018

Example

1 + 30*x^2 + 3870*x^4 + 698880*x^6 + ...

Maple

with(combinat):
seq(add(binomial(3n+k, k)*binomial(n, k)*(3n)!/n!^3, k = 0..n), n = 0..20); # Peter Bala, Jan 27 2018

Mathematica

a[n_] := (3n)! Hypergeometric2F1[-n, 3n + 1, 1, -1]/n!^3;
a /@ Range[0, 20] (* Jean-François Alcover, Feb 29 2020, after Peter Bala *)

Prog

(PARI)
my(x='x, y='y, z='z, w='w);
R = 1/(1-(w*x*z+x*y*z+w*y+x*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)));
  return(a);
};
diag(23, R, [x, y, z, w])

Crossrefs

Cf. A268545-A268555.

Sequence in context: A159993 A153470 A289247 * A127849 A166845 A166834

Adjacent sequences: A275046 A275047 A275048 * A275050 A275051 A275052

Keyword

nonn,easy

Author

Gheorghe Coserea, Jul 19 2016

Status

approved