A002754 Related to coefficient of m in Jacobi elliptic function cn(z, m).

0, 0, 4, 44, 408, 3688, 33212, 298932, 2690416, 24213776, 217924020, 1961316220, 17651846024, 158866614264, 1429799528428, 12868195755908, 115813761803232, 1042323856229152, 9380914706062436, 84428232354561996, 759854091191058040

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• A. Cayley, An Elementary Treatise on Elliptic Functions. Bell, London, 1895, p. 56.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• 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 = 0..300
• Roland Bacher and Philippe Flajolet, Pseudo-factorials, elliptic functions, and continued fractions, arXiv:0901.1379 [math.CA], 2009.
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• A. Cayley, An Elementary Treatise on Elliptic Functions (page images), G. Bell and Sons, London, 1895, p. 56.
• 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.
• J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 92.
• Index entries for linear recurrences with constant coefficients, signature (11,-19,9).

Crossrefs

Cf. A060627, A014832.

Programs

Magma

[(9^n-8*n-1)/16: n in [0..25]]; // Vincenzo Librandi, Jun 29 2011

Mathematica

a[ n_] := If[ n < 0, 0, (-1)^n (2 n)! Coefficient[ SeriesCoefficient[ JacobiCN[x, m], {x, 0, 2 n}], m, 1]]; (* Michael Somos, Dec 27 2014 *)
LinearRecurrence[{11, -19, 9}, {0, 0, 4}, 21] (* Jean-François Alcover, Sep 21 2017 *)

PARI

{a(n) = (9^n - 8*n -1) / 16}; /* Michael Somos, Jun 27 2003 */

Formula

From Michael Somos, Jun 27 2003: (Start)

G.f.: 4*x^2/((1-x)^2*(1-9*x)).

a(n) = (9^n-8*n-1)/16. (End)

a(n+2) = 4*A014832(n+1). [Bruno Berselli, Jun 29 2011]

Extensions

More terms from Paolo Dominici (pl.dm(AT)libero.it) using formulas 16.22.1 and 16.22.2 of Abramowitz and Stegun's Handbook of Mathematical Functions.