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 A060627 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). 11
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 Table of n, a(n) for n=1..36. 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 Row sums: A000364. Cf. A002754, A060628, A145271, A181612, A181613. Sequence in context: A373459 A269906 A092667 * A113101 A365088 A372013 Adjacent sequences: A060624 A060625 A060626 * A060628 A060629 A060630 KEYWORD easy,nonn,tabl AUTHOR Vladeta Jovovic, Apr 13 2001 STATUS approved

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