OFFSET
1,3
COMMENTS
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
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
REFERENCES
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 526.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(5.2.20).
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 374.
LINKS
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.22.2.
P. Bala, A triangle for calculating A060627
F. Clarke, The Taylor Series Coefficients of the Jacobi Elliptic Functions, slides. [broken link]
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052 [math.CA], 2005.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Erster Teil, p. 399 with p. 397, Springer, Berlin, Heidelberg, 2012.
Eric W. Weisstein, Jacobi Elliptic Functions
FORMULA
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).
From Peter Bala, Aug 23 2011: (Start)
The Taylor expansion of the Jacobian elliptic function cn(x,k) begins
cn(x,k) = 1 - x^2/2! + (1+4*k^2)*x^4/4! - (1+44*k^2+16*k^4)*x^6/6! + ....
The coefficient polynomials in this expansion can be calculated using nested derivatives as follows (see [Dominici, Theorem 4.1 and Example 4.5]):
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.
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).
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).
(End)
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
EXAMPLE
The first rows of triangle T(n, k), n >= 1, k = 0..n-1, are:
[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], ...
MAPLE
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
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
AUTHOR
Vladeta Jovovic, Apr 13 2001
STATUS
approved