cp's OEIS Frontend

A275051 Expansion of 3F2([1/9, 4/9, 5/9], [1/3,1], 729*x).

%I A275051 #36 Apr 11 2023 17:09:40
%S A275051 1,60,20475,9373650,4881796920,2734407111744,1605040007778900,
%T A275051 973419698810097000,604759111060745718000,382741738086972337402560,
%U A275051 245810413547242455520545552,159759730493918131135425965280,104861901534978616465850670348000
%N A275051 Expansion of 3F2([1/9, 4/9, 5/9], [1/3,1], 729*x).
%C A275051 "One may consider the following conjecture: all the irreducible factors of the minimal order linear differential operator annihilating a diagonal of a rational function should be homomorphic to their adjoint (possibly on an algebraic extension). [...]
%C A275051 "If our conjecture above was correct, this would be a way to show that the series cannot be the diagonal of a rational function." (See Boukraa link.)
%H A275051 Gheorghe Coserea, <a href="/A275051/b275051.txt">Table of n, a(n) for n = 0..200</a>
%H A275051 A. Bostan, S. Boukraa, G. Christol, S. Hassani, and J-M. Maillard <a href="http://arxiv.org/abs/1211.6031">Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity</a>, arXiv:1211.6031 [math-ph], 2012.
%H A275051 S. Boukraa, S. Hassani, J-M. Maillard, and J-A. Weil, <a href="https://arxiv.org/abs/1311.2470">Differential algebra on lattice Green functions and Calabi-Yau operators (unabridged version)</a>, arXiv:1311.2470 [math-ph], 2013.
%F A275051 G.f.: hypergeom([1/9, 4/9, 5/9], [1/3,1], 729*x).
%F A275051 From _Vaclav Kotesovec_, Jul 28 2016: (Start)
%F A275051 Recurrence: n^2*(3*n - 2)*a(n) = 3*(9*n - 8)*(9*n - 5)*(9*n - 4)*a(n-1).
%F A275051 a(n) ~ Gamma(1/3) * cos(Pi/18) * 3^(6*n) / (Pi * Gamma(1/9) * n^(11/9)).
%F A275051 (End)
%F A275051 a(n) = 729^n*cos(Pi/18)*Gamma(1/3)*Gamma(1/9+n)*Gamma(4/9+n)*Gamma(5/9+n) /(Pi*Gamma(1/9)*Gamma(1/3+n)*n!^2). - _Benedict W. J. Irwin_, Aug 05 2016
%F A275051 From _Karol A. Penson_, Apr 11 2023: (Start)
%F A275051 a(n) = Integral_{x=0..729} x^n*W(x), where
%F A275051 W(x) = W1(x) + W2(x) + W3(x), and
%F A275051 W1(x) = (2*cos(Pi/18)*3^(1/3)*2^(4/9)*sqrt(Pi)*Gamma(13/18)*hypergeom([1/9, 1/9, 7/9], [5/9, 2/3], x/729))/(9*Gamma(2/3)^2*Gamma(1/9)*Gamma(8/9)^2*x^(8/9));
%F A275051 W2(x) = cos(Pi/18)*2^(1/9)*Gamma(2/9)*Gamma(1/18)*hypergeom([4/9, 4/9, 10/9], [8/9, 4/3], x/729)/(162*Gamma(2/3)*Gamma(1/9)*Pi^(3/2)*x^(5/9));
%F A275051 W3(x) = cos(Pi/18)*3^(1/6)*2^(4/9)*Gamma(5/18)*Gamma(-1/18)*hypergeom([5/9, 5/9, 11/9], [10/9, 13/9], x/729)/(324*Gamma(2/3)*Gamma(1/9)*Pi*Gamma(4/9)*x^(4/9)).
%F A275051 This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with the singularity x^(-4/9), and for x > 0 is monotonically decreasing to zero at x = 729. At x = 729  the first derivative of W(x) is infinite. (End)
%e A275051 1 + 60*x + 20475*x^2 + 9373650*x^3 + ...
%t A275051 CoefficientList[Series[HypergeometricPFQ[{1/9, 4/9, 5/9}, {1/3,1}, 729*x], {x, 0, 15}], x] (* _Vaclav Kotesovec_, Jul 28 2016 *)
%t A275051 a[n_] := FullSimplify[(729^n Cos[Pi/18] Gamma[1/3] Gamma[1/9 + n] Gamma[4/9 + n] Gamma[5/9 + n])/(Pi Gamma[1/9] Gamma[1/3 + n] n!^2)] (* _Benedict W. J. Irwin_, Aug 05 2016 *)
%o A275051 (PARI)
%o A275051 \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
%o A275051 read("hypergeom.gpi");
%o A275051 N = 21; x = 'x + O('x^N);
%o A275051 Vec(hypergeom_sym([1/9, 4/9, 5/9], [1/3,1], 729*x, N))
%o A275051 (PARI) my(x = 'x + O('x^20)); Vec(hypergeom([1/9, 4/9, 5/9], [1/3,1], 729*x)) \\ _Michel Marcus_, Apr 11 2023
%Y A275051 Cf. A268545-A268555.
%K A275051 nonn
%O A275051 0,2
%A A275051 _Gheorghe Coserea_, Jul 19 2016