A004005 Coefficients of elliptic function sn.
1, 135, 5478, 165826, 4494351, 116294673, 2949965020, 74197080276, 1859539731885, 46535238000235, 1163848723925346, 29100851707716150, 727566807977891803, 18189614152200873621, 454744658216502193656
Offset: 2
References
- I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(5.2.24).
- 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 = 2..100
- A. Cayley, An Elementary Treatise on Elliptic Functions (page images), G. Bell and Sons, London, 1895, p. 56.
- A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.
- C. L. Mallows, Letter to N. J. A. Sloane, May 16 1973
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.
- G. Viennot, Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi, J. Combin. Theory, A 29 (1980), 121-133.
Crossrefs
Leading terms in rows of triangle in A060628.
Programs
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Maple
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. A004005 := proc(n) A060628(n,2) ; end proc: seq(A004005(n),n=2..40) ; # R. J. Mathar, Jan 30 2011
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Mathematica
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 *)
Formula
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
Extensions
More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003