A004005 - OEIS

%I M5397 #45 Apr 13 2022 13:25:17

%S 1,135,5478,165826,4494351,116294673,2949965020,74197080276,

%T 1859539731885,46535238000235,1163848723925346,29100851707716150,

%U 727566807977891803,18189614152200873621,454744658216502193656

%N Coefficients of elliptic function sn.

%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(5.2.24).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A004005/b004005.txt">Table of n, a(n) for n = 2..100</a>

%H A. Cayley, <a href="http://cdl.library.cornell.edu/Hunter/hunter.pl?handle=cornell.library.math/Cayl005&amp;id=7">An Elementary Treatise on Elliptic Functions</a> (page images), G. Bell and Sons, London, 1895, p. 56.

%H A. Fransen, <a href="http://dx.doi.org/10.1090/S0025-5718-1981-0628708-X">Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k)</a>, Math. Comp., 37 (1981), 475-497.

%H C. L. Mallows, <a href="/A004004/a004004.pdf">Letter to N. J. A. Sloane, May 16 1973</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H J. Tannery and J. Molk, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k99586x/f104.image">Eléments de la Théorie des Fonctions Elliptiques (Vol. 4)</a>, Gauthier-Villars, Paris, 1902, p. 92.

%H G. Viennot, <a href="http://dx.doi.org/10.1016/0097-3165(80)90001-1">Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi</a>, J. Combin. Theory, A 29 (1980), 121-133.

%F a(n) = (5^(2*n+1) - (8*n-4)*3^(2*n+1) + 32*n^2 - 32*n -17)/256. - _Vaclav Kotesovec_ after Fransen, Jul 30 2013

%p A004005:=-(-1-89*z+69*z**2+405*z**3)/(-1+25*z)/(9*z-1)**2/(z-1)**3; # Conjectured by _Simon Plouffe_ in his 1992 dissertation.

%p A004005 := proc(n) A060628(n,2) ; end proc: seq(A004005(n),n=2..40) ; # _R. J. Mathar_, Jan 30 2011

%t maxn = 16; se = Series[JacobiSN[u, m], {u, 0, 2*maxn+1}]; cc = Partition[CoefficientList[se, u], 2][[All, 2]]; cc2 = (CoefficientList[#, m] & /@ cc)*Table[(-1)^n*(2*n+1)!, {n, 0, maxn}]; Table[cc2[[n+1, n-1]], {n, 2, maxn}](* _Jean-François Alcover_, Feb 17 2012 *)

%Y Leading terms in rows of triangle in A060628.

%K nonn,nice,easy

%O 2,2

%A _N. J. A. Sloane_, _Simon Plouffe_

%E More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003