A002639 Numerators of expansion of Jacobi nome q in parameter m.

0, 1, 1, 21, 31, 6257, 10293, 279025, 483127, 435506703, 776957575, 22417045555, 40784671953, 9569130097211, 17652604545791, 523910972020563, 976501268709949, 935823746406530603, 1758220447807291611

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• frac

Examples

			q = 1/16*m + 1/32*m^2 + 21/1024*m^3 + 31/2048*m^4 + 6257/524288*m^5 + ...
		

References

• Guide to Tables, Math. Tables Other Aids Computation, 3 (1948), Section III, p. 234.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• Charles Hermite, Oeuvres.
• Charles Hermite, Sur Quelques Développements En Série de la Théorie Des Fonctions Elliptiques , Oeuvres. Vol. 4, Gauthier-Villars, Paris, 1917, p. 477.
• Charles Hermite, Annotated scan of a page from the Oeuvres, together with a page from Math. Tables Aids Comp., Vol. 3, 1948 that refers to it.
• J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 141.
• Eric Weisstein's World of Mathematics, Nome
• Robert M. Ziff, On Cardy's formula for the critical crossing probability in 2d percolation, J. Phys. A. 28, 1249-1255 (1995).

Crossrefs

Cf. A119349 (denominators), A002103 (where there are further references), A005797.

Programs

Maple

A002639 := proc(n::integer)
    local z;
    # coeftayl(EllipticNome(z),z=0,n) ; # very slow
    taylor(EllipticNome(z),z=0,2*n+1) ;
    convert(%,polynom) ;
    coeff(%,z,2*n) ;
    numer(%) ;
end proc:
seq(A002639(n),n=0..10) ; # R. J. Mathar, Mar 26 2025

Mathematica

Numerator[ CoefficientList[ Series[ EllipticNomeQ[m], {m, 0, 18}], m]] (* Jean-François Alcover, Sep 21 2011 *)

PARI

{a(n) = if( n<1, 0, numerator( polcoeff( serreverse( x * prod(k=1, n, (1 + x^k)^(-1)^k, 1 +x * O(x^n))^8), n) / 4^n))}

Extensions

Edited by Michael Somos, Aug 09 2002