A274789 - OEIS

%I #24 Jun 27 2023 09:21:00

%S 1,9,241,9129,402321,19321689,981044401,51794295849,2814649754641,

%T 156399050208729,8845463571211521,507517525088436729,

%U 29468616564702121041,1728353228376135226329,102242911938342248555121,6093340217607472063134249,365501683327659682186607121,22049503920365906645420399769

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

%H Vaclav Kotesovec, <a href="/A274789/b274789.txt">Table of n, a(n) for n = 0..548</a>

%H A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, 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>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendreTransform.html">Legendre Transform</a>

%F 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.

%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)*(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).

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

%F From _Peter Bala_, Jun 26 2023: (Start)

%F 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.

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

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

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

%t 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}];

%t Table[a[n], {n, 0, 17}] (* _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+w*z+x*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. A002894, A243945, A268545-A268555.

%K nonn,easy

%O 0,2

%A _Gheorghe Coserea_, Jul 14 2016