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%I #38 Jan 22 2020 03:29:58
%S 1,1,4,1,44,16,1,408,912,64,1,3688,30768,15808,256,1,33212,870640,
%T 1538560,259328,1024,1,298932,22945056,106923008,65008896,4180992,
%U 4096,1,2690416,586629984,6337665152,9860488448,2536974336,67047424,16384
%N 1 + Sum_{n >= 1} Sum_{k = 0..n-1} (-1)^n*T(n,k)*y^(2*k)*x^(2*n)/(2*n)! = JacobiCN(x,y).
%C Essentially same triangle as triangle given by [1, 0, 9, 0, 25, 0, 49, 0, 81, 0, 121, ...] DELTA [0, 4, 0, 16, 0, 36, 0, 64, 0, 100, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jun 13 2004
%C For the recurrence of the row polynomials b_n(y^2) for cn(x|y^2) = Sum_{n >=0} b_n(y^2)*x^(2*n)/(2*n)! see the Fricke reference, where y=k. - _Wolfdieter Lang_, Jul 05 2016
%D CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 526.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(5.2.20).
%D H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://people.math.sfu.ca/~cbm/aands/toc.htm">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.22.2.
%H P. Bala, <a href="/A060627/a060627.pdf">A triangle for calculating A060627</a>
%H F. Clarke, <a href="http://maths.swan.ac.uk/staff/fwc/Gregynog-slides.pdf">The Taylor Series Coefficients of the Jacobi Elliptic Functions</a>, slides. [broken link]
%H D. Dominici, <a href="http://arxiv.org/abs/math/0501052">Nested derivatives: A simple method for computing series expansions of inverse functions.</a> arXiv:math/0501052 [math.CA], 2005.
%H R. Fricke, <a href="https://doi.org/10.1007/978-3-642-19557-0_8">Die elliptischen Funktionen und ihre Anwendungen, Erster Teil</a>, p. 399 with p. 397, Springer, Berlin, Heidelberg, 2012.
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/JacobiEllipticFunctions.html">Jacobi Elliptic Functions</a>
%F JacobiCN(x, y) = 1 - 1/2*x^2 + (1/24 + 1/6*y^2)*x^4 + ( - 1/720 - 11/180*y^2 - 1/45*y^4)*x^6 + (1/40320 + 17/1680*y^2 + 19/840*y^4 + 1/630*y^6)*x^8 + ( - 1/3628800 - 247/56700*y^6 - 461/453600*y^2 - 641/75600*y^4 - 1/14175*y^8)*x^10 + O(x^12).
%F From _Peter Bala_, Aug 23 2011: (Start)
%F The Taylor expansion of the Jacobian elliptic function cn(x,k) begins
%F cn(x,k) = 1 - x^2/2! + (1+4*k^2)*x^4/4! - (1+44*k^2+16*k^4)*x^6/6! + ....
%F The coefficient polynomials in this expansion can be calculated using nested derivatives as follows (see [Dominici, Theorem 4.1 and Example 4.5]):
%F Let f(x) = sqrt(k^2-sin^2(x)). Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0.
%F Then the coefficient polynomial R(2*n,k) of x^(2*n)/(2*n)! in the expansion of cn(x,k) is given by R(2*n,k) = D^(2*n)[f](0).
%F See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x). See A181613 for the expansion of the reciprocal function 1/cn(x,k).
%F (End)
%F G.f. 1/(1 - x/(1 - (2*k)^2*x/(1 - 3^2*x/(1 - (4*k)^2*x/(1 - 5^2*x/(1 - ...)))))) = 1 + x + (1 + 4*k^2)*x^2 + (1 + 44*k^2 + 16*k^4)*x^3 + ... (see Wall, 94.19, p. 374). - _Peter Bala_, Apr 25 2017
%e The first rows of triangle T(n, k), n >= 1, k = 0..n-1, are:
%e [1], [1, 4], [1, 44, 16], [1, 408, 912, 64], [1, 3688, 30768, 15808, 256], [1, 33212, 870640, 1538560, 259328, 1024], [1, 298932, 22945056, 106923008, 65008896, 4180992, 4096], [1, 2690416, 586629984, 6337665152, 9860488448, 2536974336, 67047424, 16384], ...
%p A060627 := proc(n,m) JacobiCN(z,k) ; coeftayl(%,z=0,2*n) ; (-1)^n*coeftayl(%,k=0,2*m)*(2*n)! ; end proc: # _R. J. Mathar_, Jan 30 2011
%t nmax = 8; se = Series[JacobiCN[x, y], {x, 0, 2*nmax} ]; t[n_, m_] := (-1)^n*Coefficient[se, x, 2*n] *(2*n)! // Coefficient[#, y, m]&; Table[t[n, m], {n, 1, nmax}, {m, 0, n-1}] // Flatten (* _Jean-François Alcover_, Mar 26 2013 *)
%Y Row sums: A000364.
%Y Cf. A002754, A060628, A145271, A181612, A181613.
%K easy,nonn,tabl
%O 1,3
%A _Vladeta Jovovic_, Apr 13 2001
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