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

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