A274789 Diagonal of the rational function 1/(1-(wxyz + wxy + wxz + wy + wz + xy + xz + y + z)).

1, 9, 241, 9129, 402321, 19321689, 981044401, 51794295849, 2814649754641, 156399050208729, 8845463571211521, 507517525088436729, 29468616564702121041, 1728353228376135226329, 102242911938342248555121, 6093340217607472063134249, 365501683327659682186607121, 22049503920365906645420399769

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..548

A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv:1507.03227 [math-ph], 2015.

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

Eric Weisstein's World of Mathematics, Legendre Transform

Formula

0 = (-x^2+66*x^3+x^4-132*x^5+x^6+66*x^7-x^8)*y''' + (-3*x+303*x^2-396*x^3-594*x^4+405*x^5+291*x^6-6*x^7)*y'' + (-1+224*x-937*x^2+112*x^3+665*x^4+200*x^5-7*x^6)*y' + (9-169*x+254*x^2+30*x^3+5*x^4-x^5)*y, where y is g.f.

From Vaclav Kotesovec, Mar 19 2023: (Start)

Recurrence: (n-2)*n^3*(2*n - 5)*a(n) = (2*n - 5)*(2*n - 1)*(34*n^3 - 102*n^2 + 76*n - 17)*a(n-1) - (2*n - 3)*(134*n^4 - 804*n^3 + 1606*n^2 - 1200*n + 291)*a(n-2) + (2*n - 5)*(2*n - 1)*(34*n^3 - 204*n^2 + 382*n - 211)*a(n-3) - (n-3)^3*(n-1)*(2*n - 1)*a(n-4).

a(n) ~ 17^(1/4) * (33 + 8*sqrt(17))^(n + 1/2) / (16 * Pi^(3/2) * n^(3/2)). (End)

From Peter Bala, Jun 26 2023: (Start)

a(n) = Sum_{k = 0..n} binomial(n,k)*binomial(n+k,k)*binomial(2*k,k)^2 = Sum_{k = 0..n} binomial(n+k,n-k)*binomial(2*k,k)^3, i.e., a(n) is the Legendre transform of A002894. Cf. A243945.

a(n) = hypergeom([1/2, 1/2, -n, n + 1], [1, 1, 1], -16).

O.g.f.: A(x) = Sum_{n >= 0} binomial(2*n,n)^3 * x^n / (1 - x)^(2*n+1). (End)

Maple

seq(simplify(hypergeom([1/2, 1/2, -n, n + 1], [1, 1, 1], -16)), n = 0..20); # Peter Bala, Jun 26 2023

Mathematica

a[n_] := SeriesCoefficient[1/(1 - (w x y z + w x y + w x z + w y + w z + x y + x z + y + z)), {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Nov 16 2018 *)

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+w*z+x*y+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)));
  return(a);
};
diag(12, R, [x, y, z, w])

Crossrefs

Cf. A002894, A243945, A268545-A268555.

Sequence in context: A211052 A085799 A278858 * A263158 A183903 A251670

Adjacent sequences: A274786 A274787 A274788 * A274790 A274791 A274792

Keyword

nonn,easy

Author

Gheorghe Coserea, Jul 14 2016

Status

approved