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A004005 Coefficients of elliptic function sn.
(Formerly M5397)
3
1, 135, 5478, 165826, 4494351, 116294673, 2949965020, 74197080276, 1859539731885, 46535238000235, 1163848723925346, 29100851707716150, 727566807977891803, 18189614152200873621, 454744658216502193656 (list; graph; refs; listen; history; text; internal format)
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
A. Cayley, An Elementary Treatise on Elliptic Functions (page images), G. Bell and Sons, London, 1895, p. 56.
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.
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
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 * A248009 A143404 A051028
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
STATUS
approved

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Last modified March 19 06:50 EDT 2024. Contains 370953 sequences. (Running on oeis4.)