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 A274659 #32 Aug 13 2023 17:33:03
%S A274659 1,1,1,-1,0,1,-1,-2,0,1,2,1,-2,0,1,2,3,0,-2,0,1,-4,-2,3,0,-2,0,1,-4,
%T A274659 -5,1,3,0,-2,0,1,7,3,-6,0,3,0,-2,0,1,7,9,-2,-6,0,3,0,-2,0,1,-11,-5,11,
%U A274659 1,-6,0,3,0,-2,0,1,-11,-15,3,11,0,-6,0,3,0,-2,0,1,17,9,-17,-2,11,0,-6,0,3,0,-2,0,1
%N A274659 Triangle entry T(n, m) gives the m-th contribution T(n, m)*sin((2*m+1)*v) to the coefficient of q^n in the Fourier expansion of Jacobi's elliptic sn(u|k) function when expressed in the variables v = u/(2*K(k)/Pi) and q, the Jacobi nome, written as series in (k/4)^2. K is the real quarter period of elliptic functions.
%C A274659 If one takes the row polynomials as R(n, x) = Sum_{m=0..n} T(n, m)*x^(2*m+1), n >= 0, Jacobi's elliptic sn(u|k) function in terms of the new variables v and q becomes sn(u|k) = Sum_{n>=0} R(n, x)*q^n, if one replaces in R(n, x) x^j by sin(j*v).
%C A274659 v=v(u,k^2) and q=q(k^2) are computed with the help of A038534/A056982 for (2/Pi)*K and A002103 for q expanded in powers of (k/4)^2.
%C A274659 A test for sn(u|k) with u = 1, k = sqrt(1/2), that is v approximately 0.8472130848 and q approximately 0.04321389673, with rows n=0..10 (q powers not exceeding 10) gives 0.8030018002 to be compared with sn(1|sqrt(1/2)) approximately 0.8030018249.
%C A274659 For the derivation of the Fourier series formula of sn given in Abramowitz-Stegun (but there the notation sn(u|m=k^2) is used for sn(u|k)) see, e.g., Whittaker and Watson, p. 511 or Armitage and Eberlein, Exercises on p. 55.
%C A274659 For the cn expansion see A274661.
%C A274659 See also the W. Lang link, equations (34) and (35).
%D A274659 J. V. Armitage and W. F. Eberlein, Elliptic Functions, London Mathematical Society, Student Texts 67, Cambridge University Press, 2006.
%D A274659 E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, reprinted, 1958, Cambridge at the University Press.
%H A274659 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, p. 375, 16.23.1.
%H A274659 Wolfdieter Lang, <a href="/A273506/a273506_5.pdf">Expansions for phase space coordinates for the plane pendulum</a>
%F A274659 T(n, m) = [x^(2*m+1)]Sum_{j=0..n} b(j, x)*a(n-j), with a(k) = A274621(k/2) if k is even and a(k) = 0 if k is odd, and b(j, x) = Sum_{r | 2*j+1} x^r = Sum_{k=1..A099774(j+1)} x^(A274658(j, k)), for j >= 0.
%e A274659 The triangle T(n, m) begins:
%e A274659 m 0 1 2 3 4 5 6 7 8 9 10 11
%e A274659 n\ 2m+1 1 3 5 7 9 11 13 15 17 19 21 23
%e A274659 0: 1
%e A274659 1: 1 1
%e A274659 2: -1 0 1
%e A274659 3: -1 -2 0 1
%e A274659 4: 2 1 -2 0 1
%e A274659 5: 2 3 0 -2 0 1
%e A274659 6: -4 -2 3 0 -2 0 1
%e A274659 7: -4 -5 1 3 0 -2 0 1
%e A274659 8: 7 3 -6 0 3 0 -2 0 1
%e A274659 9: 7 9 -2 -6 0 3 0 -2 0 1
%e A274659 10: -11 -5 11 1 -6 0 3 0 -2 0 1
%e A274659 11: -11 -15 3 11 0 -6 0 3 0 -2 0 1
%e A274659 ...
%e A274659 T(4, 0) = 2 from the x^1 term in b(0, x)*a(4) + b(2, x)*a(2) + b(4, x)*a(0), that is x^1*3 + x^1*(-2) + x^1*1 = +2*x^1.
%e A274659 n=4: R(4, x) = 2*x^1 + 1*x^3 - 2*x^5 + 0*x^7 + 1*x^9, that is the sn(u|k) contribution of order q^4 in the new variables v and q is (2*sin(1*v) + 1*sin(3*v) - 2*sin(5*v) + 1*sin(9*v))*q^4.
%Y A274659 Cf. A002103, A038534/A056982, A099774, A274621, A274658, A274661.
%K A274659 sign,tabl,easy
%O A274659 0,8
%A A274659 _Wolfdieter Lang_, Jul 18 2016