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A145495 Coefficients of certain power series associated with Atkin polynomials (see Kaneko-Zagier reference for precise definition).

%I A145495 #30 Aug 19 2018 09:36:10
%S A145495 1,84,27720,13693680,5354228880,2489716429200,1010824870255200,
%T A145495 459492105307435200,189737418627305920800,85223723866764909426000,
%U A145495 35532611270849849570013600,15842376246977818384652245440,6646596943618421076833646609600,2948532659526725719238433845966400
%N A145495 Coefficients of certain power series associated with Atkin polynomials (see Kaneko-Zagier reference for precise definition).
%H A145495 Seiichi Manyama, <a href="/A145495/b145495.txt">Table of n, a(n) for n = 0..379</a>
%H A145495 M. Kaneko and D. Zagier, <a href="http://www2.math.kyushu-u.ac.jp/~mkaneko/papers/atkin.pdf">Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials</a>, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998
%F A145495 From _Seiichi Manyama_, Aug 19 2018: (Start)
%F A145495 a(n) = (6*n+1)!/((n-1)!*(2*n)!*(3*n)!*(6*n+(-1)^n)) for n > 0.
%F A145495 a(n) = 12*(6*n-6+(-1)^(n-1))*(6*n+(-1)^(n-1))*a(n-1)/((n-1)*n) for n > 1. (End)
%e A145495 From _Seiichi Manyama_, Aug 19 2018: (Start)
%e A145495 Phi_0(t)/1       = 1 + 120*t +  83160*t^2 + ... (See A001421).
%e A145495 Phi_1(t)/(84*t)  = 1 + 450*t + 394680*t^2 + ... (See A145492).
%e A145495 Phi_2(t)/(27720*t^2)
%e A145495 = (1 + 450*t + 394680*t^2 + ... - (1 + 120*t +  83160*t^2 + ... ))/(330*t)
%e A145495 = 1 + 944*t + 1054170*t^2 + ... (See A145493).
%e A145495 Phi_3(t)/(13693680*t^3)
%e A145495 = (1 + 944*t + 1054170*t^2 + ... - (1 + 450*t + 394680*t^2 + ... ))/(494*t)
%e A145495 = 1 + 1335*t + 1757970*t^2 + ... (See A145494).
%e A145495 Phi_4(t)/(5354228880*t^4)
%e A145495 = (1 + 1335*t + 1757970*t^2 + ... - (1 + 944*t + 1054170*t^2 + ... ))/(391*t)
%e A145495 = 1 + 1800*t + 2783760*t^2 + ... .
%e A145495 Phi_5(t)/(2489716429200*t^5)
%e A145495 = (1 + 1800*t + 2783760*t^2 + ... - (1 + 1335*t + 1757970*t^2 + ... ))/(465*t)
%e A145495 = 1 + 2206*t + 3863952*t^2 + ... .
%e A145495 Phi_6(t)/(1010824870255200*t^6)
%e A145495 = (1 + 2206*t + 3863952*t^2 + ... - (1 + 1800*t + 2783760*t^2 + ... ))/(406*t)
%e A145495 = 1 + 18624/7*t + 36827541/7*t^2 + ... .
%e A145495 Phi_7(t)/(459492105307435200*t^6)
%e A145495 = (1 + 18624/7*t + 36827541/7*t^2 + ... - (1 + 2206*t + 3863952*t^2 + ... ))/((3182/7)*t)
%e A145495 = 1 + (6147/2)*t + 6715687*t^2 + ... . (End)
%Y A145495 Cf. A001421, A145492, A145493, A145494.
%K A145495 nonn
%O A145495 0,2
%A A145495 _N. J. A. Sloane_, Feb 28 2009
%E A145495 More terms from _Seiichi Manyama_, Aug 19 2018