This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A002639 M5108 N2212 #50 Mar 26 2025 06:41:52
%S A002639 0,1,1,21,31,6257,10293,279025,483127,435506703,776957575,22417045555,
%T A002639 40784671953,9569130097211,17652604545791,523910972020563,
%U A002639 976501268709949,935823746406530603,1758220447807291611
%N A002639 Numerators of expansion of Jacobi nome q in parameter m.
%D A002639 Guide to Tables, Math. Tables Other Aids Computation, 3 (1948), Section III, p. 234.
%D A002639 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002639 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002639 Vincenzo Librandi, <a href="/A002639/b002639.txt">Table of n, a(n) for n = 0..100</a>
%H A002639 Charles Hermite, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAS7821">Oeuvres</a>.
%H A002639 Charles Hermite, <a href="https://archive.org/details/oeuvresdecharles04hermuoft/page/477/mode/1up?view=theater">Sur Quelques Développements En Série de la Théorie Des Fonctions Elliptiques </a>, Oeuvres. Vol. 4, Gauthier-Villars, Paris, 1917, p. 477.
%H A002639 Charles Hermite, <a href="/A002639/a002639.pdf">Annotated scan of a page from the Oeuvres</a>, together with a page from Math. Tables Aids Comp., Vol. 3, 1948 that refers to it.
%H A002639 J. Tannery and J. Molk, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k99586x/f153.image">Eléments de la Théorie des Fonctions Elliptiques (Vol. 4)</a>, Gauthier-Villars, Paris, 1902, p. 141.
%H A002639 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Nome.html">Nome</a>
%H A002639 Robert M. Ziff, <a href="http://dx.doi.org/10.1088/0305-4470/28/5/013">On Cardy's formula for the critical crossing probability in 2d percolation</a>, J. Phys. A. 28, 1249-1255 (1995).
%e A002639 q = 1/16*m + 1/32*m^2 + 21/1024*m^3 + 31/2048*m^4 + 6257/524288*m^5 + ...
%p A002639 A002639 := proc(n::integer)
%p A002639 local z;
%p A002639 # coeftayl(EllipticNome(z),z=0,n) ; # very slow
%p A002639 taylor(EllipticNome(z),z=0,2*n+1) ;
%p A002639 convert(%,polynom) ;
%p A002639 coeff(%,z,2*n) ;
%p A002639 numer(%) ;
%p A002639 end proc:
%p A002639 seq(A002639(n),n=0..10) ; # _R. J. Mathar_, Mar 26 2025
%t A002639 Numerator[ CoefficientList[ Series[ EllipticNomeQ[m], {m, 0, 18}], m]] (* _Jean-François Alcover_, Sep 21 2011 *)
%o A002639 (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))}
%Y A002639 Cf. A119349 (denominators), A002103 (where there are further references), A005797.
%K A002639 nonn,nice,frac
%O A002639 0,4
%A A002639 _N. J. A. Sloane_
%E A002639 Edited by _Michael Somos_, Aug 09 2002