A275791 -id:A275791 - cp's OEIS frontend

A274661 Triangle read by rows: T(n, m) gives the m-th contribution T(n, m)cos((2m+1)v) to the coefficient of q^n in the Fourier expansion of Jacobi's elliptic cn(u|k) function when expressed in the variables v = u/(2K(k)/Pi) and q, the Jacobi nome, written as series in (k/4)^2. K is the real quarter period of elliptic functions.

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, -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, -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, -17, 23, 6, -18, -1, 11, 0, -6, 0, 3, 0, -2, 0, 1

Offset: 0

Wolfdieter Lang, Jul 27 2016

If one takes the row polynomials as P(n, x) = Sum_{m=0..n} T(n, m)*x^m, n >= 0, Jacobi's elliptic function cn(u|k) in terms of the new variables v and q becomes cn(u|k) = Sum_{n>=0} P(n, x)*q^n, if in P(n, x) one replaces x^j by cos((2*j+1)*v).
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.
A test for cn(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 0.5959766014 to be compared with cn(1|sqrt(1/2)) approximately 0.5959765676.
For the derivation of the Fourier series formula of cn 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.
For sn see A274659 (differently signed triangle).
The sum of entries in row n is P(n, 1) = A000007(n): 1, repeat 0. Proof: due to the g.f. identity (from the convolution)
  Sum_{n >= 0} x^n/(1 + x^(2*n+1))  = (Sum_{n >= 0} x^(n*(n+1)))^2.
  This is proved by bisecting the g.f. on the l.h.s. which generates c(n, 1) = (-1)^n*Sum_{2*r+1 | 2*n+1} (-1)^n. The part with n = 2*k+1 vanishes due to r_2(4*k+1)/4 = 0, where r_2(n) is the number of solutions of n as a sum of two squares. See the Grosswald reference. The part with n = 2*k becomes Sum_{k >= 0} x^(2*k) r_2(4*k+1)/4 which is the r.h.s. See A008441, the Broadhurst Oct 20 2002 comment.
For another version of this expansion of cn see A275791.
See also the W. Lang link, eqs. (43) and (44). - Wolfdieter Lang, Aug 26 2016

			The triangle T(n, m) begins:
      m  0   1  2  3  4  5  6  7  8  9 10 11
n\ 2m+1  1   3  5  7  9 11 13 15 17 19 21 23
0:       1
1:      -1   1
2:      -1   0  1
3:       1  -2  0  1
4:       2  -1 -2  0  1
5:      -2   3  0 -2  0  1
6:      -4   2  3  0 -2  0  1
7:       4  -5 -1  3  0 -2  0  1
8:       7  -3 -6  0  3  0 -2  0  1
9:      -7   9  2 -6  0  3  0 -2  0  1
10:    -11   5 11 -1 -6  0  3  0 -2  0  1
11:     11 -15 -3 11  0 -6  0  3  0 -2  0  1
...
n = 4: c(0, x)*a(4) + c(2, x)*a(2) + c(4, x)*a(0) = (+x^1)*3 +  (+x^1 + x^5)*(-2) + (+x^1 - x^3 + x^9)*1 = +2*x^1 - x^3 - 2*x^5 + 0*x^7 + x^9. Hence row n=4 is 2, -1, -2, 0, 1.
From A274660, row n = 4: c(4, x) = +x^1 - x^3 +x^9.
n = 4: P(4, x) = 2 - 1*x^1 - 2*x^2 + 1*x^4, that is the contribution of order q^4 to cn in the new variables is (2*cos(v)  - 1*cos(3*v) - 2*cos(5*v) + 1*cos(9*v))*q^4.

J. V. Armitage and W. F. Eberlein, Elliptic Functions, London Mathematical Society, Student Texts 67, Cambridge University Press, 2006.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15, Theorem 3.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, reprinted, 1958, Cambridge at the University Press.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 375, 16.23.2.
Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum

Cf. A000007, A099774, A274621, A274659, A274660, A275791.

T(n, m) = [x^(2*m+1)]Sum_{j=0..n} c(j, x)*a(n-j), with a(k) = A274621(k/2) if k is even and a(k) = 0 if k is odd, and c(j, x) = (-1)^j*Sum_{2*r+1 | 2*j+1} (-1)^r*x^(2*r+1) = Sum_{k=1..A099774(j+1)} sign(A274660(j, k))*x^(abs(A274660(j, k))), for j >= 0.

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