A145093 Constant term in Atkin polynomial A_n(j).

1, -720, 269280, -107765856, 44184000960, -18343724398560, 7674347243833920, -3227358183233849280, 1362313994259911121792, -576679534187816788835040, 244653763082978694519455040, -103977849310265945170768392000, 44255109760585207541022458448000, -18858872473375780341531310443030720

Offset: 0

Author

N. J. A. Sloane, Feb 28 2009

Keywords

• sign

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

Programs

Maple

af:=proc(a,n) mul(a+i,i=0..n-1); end; A0:=n->(-12)^(3*n+1)*af(-1/12,n)*af(5/12,n)/(2*n-1)!;

Mathematica

Flatten[{1, Table[FullSimplify[(-1)^n * 2^(4*n + 1) * 3^(3*n) * Gamma[2*n - 1/6] / (Gamma[5/6] * Gamma[2*n])], {n, 1, 20}]}] (* Vaclav Kotesovec, Apr 07 2018 *)

Formula

Theorem 4 on page 100 and Proposition 6 on page 117 of the Kaneko-Zagier reference gives an explicit formula and a recurrence for these polynomials. See Maple code.

From Vaclav Kotesovec, Apr 07 2018: (Start)

For n > 0, a(n) = (-1)^n * 2^(4*n + 1) * 3^(3*n) * Gamma(2*n - 1/6) / (Gamma(5/6) * Gamma(2*n)).

a(n) ~ (-1)^n * 2^(4*n + 5/6) * 3^(3*n) / (Gamma(5/6) * n^(1/6)). (End)