A002754 Related to coefficient of m in Jacobi elliptic function cn(z, m). (Formerly M3680 N1501)

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

Offset

0,3

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).

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]

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 *)

Prog

(PARI) {a(n) = (9^n - 8*n -1) / 16}; /* Michael Somos, Jun 27 2003 */
(Magma) [(9^n-8*n-1)/16: n in [0..25]]; // Vincenzo Librandi, Jun 29 2011

Crossrefs

Cf. A060627, A014832.

Sequence in context: A178294 A043039 A198962 * A187870 A216272 A221405

Adjacent sequences: A002751 A002752 A002753 * A002755 A002756 A002757

Keyword

nonn,easy

Author

N. J. A. Sloane

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.

Status

approved