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 A145495 Coefficients of certain power series associated with Atkin polynomials (see Kaneko-Zagier reference for precise definition). 1
 1, 84, 27720, 13693680, 5354228880, 2489716429200, 1010824870255200, 459492105307435200, 189737418627305920800, 85223723866764909426000, 35532611270849849570013600, 15842376246977818384652245440, 6646596943618421076833646609600, 2948532659526725719238433845966400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..379 M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998 FORMULA From Seiichi Manyama, Aug 19 2018: (Start) a(n) = (6*n+1)!/((n-1)!*(2*n)!*(3*n)!*(6*n+(-1)^n)) for n > 0. a(n) = 12*(6*n-6+(-1)^(n-1))*(6*n+(-1)^(n-1))*a(n-1)/((n-1)*n) for n > 1. (End) EXAMPLE From Seiichi Manyama, Aug 19 2018: (Start) Phi_0(t)/1 = 1 + 120*t + 83160*t^2 + ... (See A001421). Phi_1(t)/(84*t) = 1 + 450*t + 394680*t^2 + ... (See A145492). Phi_2(t)/(27720*t^2) = (1 + 450*t + 394680*t^2 + ... - (1 + 120*t + 83160*t^2 + ... ))/(330*t) = 1 + 944*t + 1054170*t^2 + ... (See A145493). Phi_3(t)/(13693680*t^3) = (1 + 944*t + 1054170*t^2 + ... - (1 + 450*t + 394680*t^2 + ... ))/(494*t) = 1 + 1335*t + 1757970*t^2 + ... (See A145494). Phi_4(t)/(5354228880*t^4) = (1 + 1335*t + 1757970*t^2 + ... - (1 + 944*t + 1054170*t^2 + ... ))/(391*t) = 1 + 1800*t + 2783760*t^2 + ... . Phi_5(t)/(2489716429200*t^5) = (1 + 1800*t + 2783760*t^2 + ... - (1 + 1335*t + 1757970*t^2 + ... ))/(465*t) = 1 + 2206*t + 3863952*t^2 + ... . Phi_6(t)/(1010824870255200*t^6) = (1 + 2206*t + 3863952*t^2 + ... - (1 + 1800*t + 2783760*t^2 + ... ))/(406*t) = 1 + 18624/7*t + 36827541/7*t^2 + ... . Phi_7(t)/(459492105307435200*t^6) = (1 + 18624/7*t + 36827541/7*t^2 + ... - (1 + 2206*t + 3863952*t^2 + ... ))/((3182/7)*t) = 1 + (6147/2)*t + 6715687*t^2 + ... . (End) CROSSREFS Cf. A001421, A145492, A145493, A145494. Sequence in context: A289325 A202923 A232914 * A275452 A269933 A184126 Adjacent sequences: A145492 A145493 A145494 * A145496 A145497 A145498 KEYWORD nonn AUTHOR N. J. A. Sloane, Feb 28 2009 EXTENSIONS More terms from Seiichi Manyama, Aug 19 2018 STATUS approved

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Last modified March 27 05:46 EDT 2023. Contains 361554 sequences. (Running on oeis4.)