A030185 Coefficients in expansion of E_2*E_4/(E_6*j) in powers of 1/j.

1, 720, 911520, 1301011200, 1958042030400, 3036508587993600, 4800606078996518400, 7692266556998230118400, 12448466349673022940816000, 20299324957420186505619072000, 33301542920000564787678367257600

Offset

1,2

References

R. A. Rankin, The zeros of Eisenstein series, Publ. Ramanujan Institute 1 (1969), 137-144. (On page 139).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Oscar E. González, An observation of Rankin on Hankel determinants, Department of Mathematics, University of Illinois at Urbana-Champaign, 2018.

M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, AMS/IP Studies in Advanced Mathematics, vol. 7, 97--126, (1998). See esp. p. 110.

Formula

a(n) ~ Pi^(3/2) * 1728^n / (72 * Gamma(1/4)^4 * sqrt(3*n)). - Vaclav Kotesovec, Apr 07 2018

Mathematica

nmax = 20; E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, nmax + 1}] + O[x]^(nmax + 1); A000521x = Normal[Series[(1728 E4[x]^3/(E4[x]^3 - E6[x]^2)), {x, 0, nmax}]]; expansion = CoefficientList[Series[E2[x]*E4[x]/(E6[x]*(1728 E4[x]^3/(E4[x]^3 - E6[x]^2))), {x, 0, nmax}], x]; A[x_] := Sum[c[k]/A000521x^k, {k, 0, nmax}]; Array[c, nmax] /. Solve[CoefficientList[Series[A[x], {x, 0, nmax}], x] == expansion][[1]] (* Vaclav Kotesovec, Apr 07 2018 *)

Crossrefs

Cf. A000521, A006352, A004009, A013973, A145200.

Sequence in context: A378468 A221621 A054779 * A227668 A010799 A283830

Adjacent sequences: A030182 A030183 A030184 * A030186 A030187 A030188

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Sep 25 2010

Status

approved