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%I M5108 N2212 #47 Jan 04 2024 14:30:55
%S 0,1,1,21,31,6257,10293,279025,483127,435506703,776957575,22417045555,
%T 40784671953,9569130097211,17652604545791,523910972020563,
%U 976501268709949,935823746406530603,1758220447807291611
%N Numerators of expansion of Jacobi nome q in parameter m.
%D Guide to Tables, Math. Tables Other Aids Computation, 3 (1948), Section III, p. 234.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A002639/b002639.txt">Table of n, a(n) for n = 0..100</a>
%H Charles Hermite, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAS7821">Oeuvres</a>.
%H 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Nome.html">Nome</a>
%H 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 q = 1/16*m + 1/32*m^2 + 21/1024*m^3 + 31/2048*m^4 + 6257/524288*m^5 + ...
%t Numerator[ CoefficientList[ Series[ EllipticNomeQ[m], {m, 0, 18}], m]] (* _Jean-François Alcover_, Sep 21 2011 *)
%o (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 Cf. A119349 (denominators), A002103 (where there are further references), A005797.
%K nonn,nice,frac
%O 0,4
%A _N. J. A. Sloane_
%E Edited by _Michael Somos_, Aug 09 2002
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