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A275051 G.f.: 3F2([1/9, 4/9, 5/9], [1/3,1], 729*x). 11
1, 60, 20475, 9373650, 4881796920, 2734407111744, 1605040007778900, 973419698810097000, 604759111060745718000, 382741738086972337402560, 245810413547242455520545552, 159759730493918131135425965280, 104861901534978616465850670348000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

"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). [...]

"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.)

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 0..200

A. Bostan, S. Boukraa, G. Christol, S. Hassani, J-M. Maillard Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity, arXiv:1211.6031 [math-ph], 2012.

S. Boukraa, S. Hassani, J-M. Maillard, J-A. Weil, Differential algebra on lattice Green functions and Calabi-Yau operators (unabridged version), arXiv:1311.2470 [math-ph], 2013.

FORMULA

G.f.: hypergeom([1/9, 4/9, 5/9], [1/3,1], 729*x).

From Vaclav Kotesovec, Jul 28 2016: (Start)

Recurrence: n^2*(3*n - 2)*a(n) = 3*(9*n - 8)*(9*n - 5)*(9*n - 4)*a(n-1).

a(n) ~ Gamma(1/3) * cos(Pi/18) * 3^(6*n) / (Pi * Gamma(1/9) * n^(11/9)).

(End)

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

EXAMPLE

1 + 60*x + 20475*x^2 + 9373650*x^3 + ...

MATHEMATICA

CoefficientList[Series[HypergeometricPFQ[{1/9, 4/9, 5/9}, {1/3, 1}, 729*x], {x, 0, 15}], x] (* Vaclav Kotesovec, Jul 28 2016 *)

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 *)

PROG

(PARI)

\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");

read("hypergeom.gpi");

N = 21; x = 'x + O('x^N);

Vec(hypergeom([1/9, 4/9, 5/9], [1/3, 1], 729*x, N))

CROSSREFS

Cf. A268545-A268555.

Sequence in context: A291912 A001460 A003794 * A068295 A177643 A123482

Adjacent sequences:  A275048 A275049 A275050 * A275052 A275053 A275054

KEYWORD

nonn

AUTHOR

Gheorghe Coserea, Jul 19 2016

STATUS

approved

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Last modified September 17 01:53 EDT 2021. Contains 347478 sequences. (Running on oeis4.)