A004005 Coefficients of elliptic function sn. (Formerly M5397)

1, 135, 5478, 165826, 4494351, 116294673, 2949965020, 74197080276, 1859539731885, 46535238000235, 1163848723925346, 29100851707716150, 727566807977891803, 18189614152200873621, 454744658216502193656, 11368657974646161302248, 284216848055029040209305, 7105425014717554019615631

Offset

2,2

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.

Arne Fransén, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(x,k), Math. Comp., 37 (1981), 475-494.

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

Jules Tannery and Jules Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.

Gérard Viennot, Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi, J. Combin. Theory, A 29 (1980), 121-133.

Index entries for linear recurrences with constant coefficients, signature (46,-663,3748,-7711,6606,-2025).

Formula

a(n) = (5^(2*n+1) - (8*n-4)*3^(2*n+1) + 32*n^2 - 32*n - 17)/256. - Vaclav Kotesovec after Fransén, Jul 30 2013

G.f.: x^2*(1 + 89*x - 69*x^2 - 405*x^3)/((1 - x)^3*(1 - 9*x)^2*(1 - 25*x)). - Andrew Howroyd, Nov 01 2025

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

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 *)

Crossrefs

Leading terms in rows of triangle in A060628.

Sequence in context: A332428 A212608 A212611 * A381258 A248009 A143404

Adjacent sequences: A004002 A004003 A004004 * A004006 A004007 A004008

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Simon Plouffe

Extensions

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

Status

approved