A145495 - OEIS

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A145495

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

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) = (6n+1)!/((n-1)!(2n)!(3n)!(6n+(-1)^n)) for n > 0.` `a(n) = 12(6n-6+(-1)^(n-1))(6n+(-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 + 120t + 83160t^2 + ... (See A001421).` `Phi_1(t)/(84t) = 1 + 450t + 394680t^2 + ... (See A145492).` `Phi_2(t)/(27720t^2)` `= (1 + 450t + 394680t^2 + ... - (1 + 120t + 83160t^2 + ... ))/(330t)` `= 1 + 944t + 1054170t^2 + ... (See A145493).` `Phi_3(t)/(13693680t^3)` `= (1 + 944t + 1054170t^2 + ... - (1 + 450t + 394680t^2 + ... ))/(494t)` `= 1 + 1335t + 1757970t^2 + ... (See A145494).` `Phi_4(t)/(5354228880t^4)` `= (1 + 1335t + 1757970t^2 + ... - (1 + 944t + 1054170t^2 + ... ))/(391t)` `= 1 + 1800t + 2783760t^2 + ... .` `Phi_5(t)/(2489716429200t^5)` `= (1 + 1800t + 2783760t^2 + ... - (1 + 1335t + 1757970t^2 + ... ))/(465t)` `= 1 + 2206t + 3863952t^2 + ... .` `Phi_6(t)/(1010824870255200t^6)` `= (1 + 2206t + 3863952t^2 + ... - (1 + 1800t + 2783760t^2 + ... ))/(406t)` `= 1 + 18624/7t + 36827541/7t^2 + ... .` `Phi_7(t)/(459492105307435200t^6)` `= (1 + 18624/7t + 36827541/7t^2 + ... - (1 + 2206t + 3863952t^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.)