|
%I #10 Aug 20 2018 08:38:27
%S 1,-720,269280,-107765856,44184000960,-18343724398560,
%T 7674347243833920,-3227358183233849280,1362313994259911121792,
%U -576679534187816788835040,244653763082978694519455040,-103977849310265945170768392000,44255109760585207541022458448000,-18858872473375780341531310443030720
%N Constant term in Atkin polynomial A_n(j).
%H Seiichi Manyama, <a href="/A145093/b145093.txt">Table of n, a(n) for n = 0..379</a>
%H 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 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.
%F From _Vaclav Kotesovec_, Apr 07 2018: (Start)
%F 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)).
%F a(n) ~ (-1)^n * 2^(4*n + 5/6) * 3^(3*n) / (Gamma(5/6) * n^(1/6)). (End)
%p 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)!;
%t 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 *)
%K sign
%O 0,2
%A _N. J. A. Sloane_, Feb 28 2009
| 