This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A275791 #20 Jan 17 2021 11:07:40
%S A275791 1,-4,1,4,-5,1,0,12,-7,1,4,-21,25,-9,1,-8,30,-63,42,-11,1,0,-44,131,
%T A275791 -138,63,-13,1,0,72,-246,365,-253,88,-15,1,4,-85,425,-837,808,-416,
%U A275791 117,-17,1,-4,85,-685,1734,-2200,1552,-635,150,-19,1,8,-134,1053,-3319,5326,-4888,2705,-918,187,-21,1
%N A275791 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function cn(u, k) divided by cos(v) in terms of the Jacobi nome q and even powers of 2*cos(v) with v = u/((2/Pi)*K(k)).
%C A275791 The representation of Jacobi's elliptic cn(u, k) function in terms of quotients of theta functions of the variables q (Jacobi nome) and v = u/((2/Pi)*K(k)) with the real quarter period K is
%C A275791 cn(u, k) = (theta_4(0, q)/theta_2(0, q)) * (theta_2(v, q)/theta_4(v, q)).
%C A275791 This can be written either in terms of infinite sums or products. (see e.g. Tricomi, p. 176, eq. (3.87), p. 156, eq. (3.51), p. 167, eq. (3.71) with (3.71'), p. 173, eq. (3.81)).
%C A275791 The sums representation involves cos((2*n+1)*v) and cos(2*n*v) functions. Using Chebyshev T polynomial (A053120) one can write cn(u, k)/cos(v) = Sum_{n >= 0} q^n*Sum_{m = 0..n} T(n, m)*(2*cos(v))^(2*m).
%C A275791 The product representation involves directly (2*cos(v))^2 powers in the q expansion:
%C A275791 cn(u, k)/cos(v) = Product_{n >= 1} ((1 - q^(2*n-1))^2 *((1 - q^(2*n))^2 + q^(2*n)*(2*cos(v))^2) / ((1 + q^(2*n))^2*((1 + q^(2*n-1))^2 - q^(2*n-1)*(2*cos(v))^2))) = Sum_{n >=0} q^n*Sum_{m = 1..n} T(n, m) * (2*cos(v))^(2*m).
%C A275791 For another version of this cn expansion see A274661.
%C A275791 For the sn(u, k)/sin(v) analog see A274662.
%C A275791 This entry is inspired by a paper of Bradley Klee giving an approximation to the phase space solution of the plane pendulum (see A273506).
%C A275791 See also the W. Lang link, equations (59) and (60).
%D A275791 F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948.
%H A275791 Wolfdieter Lang, <a href="/A273506/a273506_5.pdf">Expansions for phase space coordinates for the plane pendulum</a>
%F A275791 cn(u, k) = cos(v)*Sum_{n >= 0} q^n*Sum_{m = 0..n} T(n, m)*(2*cos(v))^(2*m), becoming an identity if q, the Jacobi nome, is replaced by exp(-Pi*K'(k)/K(k)) and v by u/((2/Pi)*K(k)) with the real and imaginary quarter periods K' and K, respectively. For the expansions of q = q(k) see A005797 or better A002103 for q = q((1-k^2)^(1/4)), and for (2/Pi)*K(k) see A038534 / A056982.
%e A275791 The triangle T(n, m) begins:
%e A275791 n\m 0 1 2 3 4 5 6 7 8 9
%e A275791 0: 1
%e A275791 1: -4 1
%e A275791 2: 4 -5 1
%e A275791 3: 0 12 -7 1
%e A275791 4: 4 -21 25 -9 1
%e A275791 5: -8 30 -63 42 -11 1
%e A275791 6: 0 -44 131 -138 63 -13 1
%e A275791 7: 0 72 -246 365 -253 88 -15 1
%e A275791 8: 4 -85 425 -837 808 -416 117 -17 1
%e A275791 9: -4 85 -685 1734 -2200 1552 -635 150 -19 1
%e A275791 ...
%e A275791 Row n=10: 8 -134 1053 -3319 5326 -4888 2705 -918 187 -21 1.
%e A275791 ...
%e A275791 n=4: q^4 term of cn(u, k)/cos(v) is 4 - 21*(2*cos(v))^2 + 25*(2*cos(v))^4 - 9*(2*cos(v))^6 + (2*cos(v))^8.
%e A275791 One can check the identity for cn(u, k), for example for u = 1 and k = sqrt(1/2), belonging to v = 0.8472130848 and q = 0.04321391815 (Maple 10 digits), with the result from Maple's cn function cn(1, sqrt(1/2)) = 0.5959765676 (10 digits). If one takes the expansion up to q^4 inclusive one obtains 0.5959776092 (10 digits). If one goes up to q^6 inclusive one gets 0.5959765640 (10 digits).
%Y A275791 Cf. A005797, A002103, A038534/A056982, A053120, A274661, A274662.
%K A275791 sign,tabl,easy
%O A275791 0,2
%A A275791 _Wolfdieter Lang_, Aug 10 2016