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A275052
G.f.: 3F2([1/7, 2/7, 4/7], [1/2, 1], 2401 x).
1
1, 112, 103488, 139087872, 219932697600, 380982080962560, 699690909055057920, 1338362619711643975680, 2637829075787918298316800, 5319794376634771700187136000, 10925401705883689450455905075200, 22771065347191895949498972005990400, 48042740185717267168321727861725593600
OFFSET
0,2
COMMENTS
"Other hypergeometric 'blind spots' for Christol’s conjecture" - (see Bostan link).
LINKS
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.
FORMULA
G.f.: hypergeom([1/7, 2/7, 4/7], [1/2, 1], 2401*x).
From Vaclav Kotesovec, Jul 28 2016: (Start)
Recurrence: n^2*(2*n - 1)*a(n) = 14*(7*n - 6)*(7*n - 5)*(7*n - 3)*a(n-1).
a(n) ~ sqrt(Pi) * 7^(4*n) / (Gamma(1/7) * Gamma(2/7) * Gamma(4/7) * n^(3/2)).
(End)
a(n) = 9604^n*Gamma(1/7+n)*Gamma(2/7+n)*Gamma(4/7+n)/(n!*(2n)!*Gamma(1/7)*Gamma(2/7)*Gamma(4/7)). - Benedict W. J. Irwin, Aug 05 2016
EXAMPLE
1 + 112*x + 103488*x^2 + 139087872*x^3 + ...
MATHEMATICA
CoefficientList[Series[HypergeometricPFQ[{1/7, 2/7, 4/7}, {1/2, 1}, 2401 x], {x, 0, 12}], x] (* Michael De Vlieger, Jul 26 2016 *)
a[n_] := FullSimplify[(9604^n Gamma[1/7 + n] Gamma[2/7 + n] Gamma[4/7 + n])/(n!(2n)! Gamma[1/7] Gamma[2/7] Gamma[4/7])] (* Benedict W. J. Irwin, Aug 05 2016 *)
PROG
(PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 12; x = 'x + O('x^N);
Vec(hypergeom([1/7, 2/7, 4/7], [1/2, 1], 2401*x, N))
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Jul 20 2016
STATUS
approved