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%I M3680 N1501 #51 Aug 23 2023 08:35:08
%S 0,0,4,44,408,3688,33212,298932,2690416,24213776,217924020,1961316220,
%T 17651846024,158866614264,1429799528428,12868195755908,
%U 115813761803232,1042323856229152,9380914706062436,84428232354561996,759854091191058040
%N Related to coefficient of m in Jacobi elliptic function cn(z, m).
%D A. Cayley, An Elementary Treatise on Elliptic Functions. Bell, London, 1895, p. 56.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A002754/b002754.txt">Table of n, a(n) for n = 0..300</a>
%H Roland Bacher and Philippe Flajolet, <a href="http://arxiv.org/abs/0901.1379">Pseudo-factorials, elliptic functions, and continued fractions</a>, arXiv:0901.1379 [math.CA], 2009.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A. Cayley, <a href="http://cdl.library.cornell.edu/Hunter/hunter.pl?handle=cornell.library.math/Cayl005&id=7">An Elementary Treatise on Elliptic Functions</a> (page images), G. Bell and Sons, London, 1895, p. 56.
%H G. Viennot, <a href="http://dx.doi.org/10.1016/0097-3165(80)90001-1">Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi</a>, J. Combin. Theory, A 29 (1980), 121-133.
%H J. Tannery and J. Molk, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k99586x/f104.image">Eléments de la Théorie des Fonctions Elliptiques (Vol. 4)</a>, Gauthier-Villars, Paris, 1902, p. 92.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-19,9).
%F From _Michael Somos_, Jun 27 2003: (Start)
%F G.f.: 4*x^2/((1-x)^2*(1-9*x)).
%F a(n) = (9^n-8*n-1)/16. (End)
%F a(n+2) = 4*A014832(n+1). [_Bruno Berselli_, Jun 29 2011]
%t 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 *)
%t LinearRecurrence[{11, -19, 9}, {0, 0, 4}, 21] (* _Jean-François Alcover_, Sep 21 2017 *)
%o (PARI) {a(n) = (9^n - 8*n -1) / 16}; /* _Michael Somos_, Jun 27 2003 */
%o (Magma) [(9^n-8*n-1)/16: n in [0..25]]; // _Vincenzo Librandi_, Jun 29 2011
%Y Cf. A060627, A014832.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E 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.
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