A056982 -id:A056982 - cp's OEIS frontend

A124399 a(n) = 4^(n - bitcount(n)) where bitcount(n) = A000120(n).

1, 1, 4, 4, 64, 64, 256, 256, 16384, 16384, 65536, 65536, 1048576, 1048576, 4194304, 4194304, 1073741824, 1073741824, 4294967296, 4294967296, 68719476736, 68719476736, 274877906944, 274877906944, 17592186044416, 17592186044416

Offset: 0

Wolfdieter Lang, Nov 10 2006

Numerators of one half of norm square of monic Legendre polynomials p_n(x).
The denominators of these polynomials are given by A069955.
The rationals N2(n) = 2*a(n)/A069955(n) give the minimal norm square for real monic polynomials. The norm square is defined as integral over the interval [-1,+1] of the square of the polynomials. Cf. the Courant-Hilbert reference.

			Rationals a(n)/A069955(n): [1, 1/3, 4/45, 4/175, 64/11025, 64/43659, 256/693693, ...].
Rationals N2(n): [2, 2/3, 8/45, 8/175, 128/11025, 128/43659, 512/693693,...].

Richard Courant and David Hilbert, Methoden der mathematischen Physik, Bd. I, 3, Auflage, Springer, 1993, pp. 73-74.

Wolfdieter Lang, Norm square, rationals and more.

Cf. A000120, A001790, A056982, A060818, A069955 (denominators of N2(n) as defined in the comments).

bitcount(n) = sum(digits(n, base=2))
a(n) = 4^(n - bitcount(n)) # Peter Luschny, Oct 01 2019

a[n_] := 4^(n - DigitCount[n, 2, 1]); Array[a, 25, 0] (* Amiram Eldar, Jul 25 2023 *)

a(n) = numerator((1/(2*n+1))*((2^n)/binomial(2*n,n))^2); \\ Michel Marcus, Aug 11 2019

a(n) = numerator(N2(n)/2) with N2(n)/2:=(1/(2*n+1))*((2^n)/binomial(2*n,n))^2.
N2(n)/2 = (1/(2*n+1))*(1/L(n))^2 with L(n)= A001790(n)/A060818(n), the leading coefficient of the Legendre polynomial P_n(x), in lowest terms.
Bisection: a(2*n)=a(2*n+1) = A056982(n), n>=0.

New name by Peter Luschny, Oct 01 2019

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.

Original entry on oeis.org

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.

A274662 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function sn(u, k) divided by sin(v) in terms of the Jacobi nome q and even powers of 2cos(v), with v = u/((2/Pi)K(k)).

Original entry on oeis.org

1, 0, 1, 0, -3, 1, 0, 4, -5, 1, 0, -3, 13, -7, 1, 0, 6, -25, 26, -9, 1, 0, -12, 43, -70, 43, -11, 1, 0, 8, -70, 157, -147, 64, -13, 1, 0, -3, 109, -315, 408, -264, 89, -15, 1, 0, 13, -155, 582, -984, 872, -429, 118, -17, 1, 0, -18, 201, -1001, 2142, -2464, 1641, -650, 151, -19, 1

Offset: 0

Wolfdieter Lang, Aug 08 2016

The representation of Jacobi's elliptic sn(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
  sn(u, k) = (theta_3(0, q)/theta_2(0, q)) * (theta_1(v, q)/theta_4(v, q)).
  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)).
The sums representation involves sin((2*n+1)*v) and cos(2*n*v) functions. Using Chebyshev S and T polynomial (A049310 and A053120) one can write sn(u, k)/sin(v) = Sum_{n >= 0} q^n*Sum_{m = 0..n} T(n, m) * (2*cos(v))^(2*m).
The product representation involves directly (2*cos(v))^2 powers in the q expansion:
  sn(u, k)/sin(v) = Product_{n >= 1} (1 - (q^(2*n)/(1 + q^(2*n))^2)*(2*cos(v))^2) / (1 - (q^(2*n-1)/(1 + q^(2*n-1))^2)*(2*cos(v))^2) = Sum_{n >=0} q^n * Sum_{m = 1..n} T(n, m)*(2*cos(v))^(2*m).
This sn expansion in the v and q variables is used in the scaled phase space coordinate qhat(v, q) of the plane pendulum. See A275790.
An alternative expansion of sn in the variables v and q is given in A274659.
See also the W. Lang link, equations (52) and (53).

			The triangle T(n, m) begins:
n\m 0   1    2    3    4    5    6   7   8 9
0:  1
1:  0   1
2:  0  -3    1
3:  0   4   -5    1
4:  0  -3   13   -7    1
5:  0   6  -25   26   -9    1
6:  0 -12   43  -70   43  -11    1
7:  0   8  -70  157 -147   64  -13   1
8:  0  -3  109 -315  408 -264   89 -15   1
9:  0  13 -155  582 -984  872 -429 118 -17 1
...
row n=10: 0 -18 201 -1001 2142 -2464 1641 -650 151 -19 1
...
n=4: the q^4 term of sn(u, k)/sin(v) is -3*(2*cos(v))^2 + 13*(2*cos(v))^4 - 7*(2*cos(v))^6 + (2*cos(v))^8.
One can check the identity 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 sn function sn(1, sqrt(1/2)) = 0.8030018249 (10 digits). If one takes the expansion up to q^4 inclusive one obtains .8030012888 (10 digits).

F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948.

Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum

Cf. A002103, A005797, A038534/A056982, A049310, A053120, A274659, A275790.

sn(u, k) = sin(v)*Sum_{n >= 0} q^n*Sum_{m = 0..n} T(n, m)*(2*cos(v))^(2*m), becoming an identity when 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.

A326748 Triangular array, read by rows: T(n,k) = denominator of Jtilde_k(n), 1 <= k <= 2*n+2.

Original entry on oeis.org

1, 1, 3, 4, 1, 1, 15, 64, 48, 8, 4, 4, 35, 256, 8640, 576, 216, 144, 36, 36, 315, 16384, 430080, 1024, 138240, 4608, 6912, 576, 576, 576, 693, 65536, 387072000, 3686400, 4838400, 30720, 576000, 115200, 43200, 11520, 14400, 14400

Offset: 0

Seiichi Manyama, Oct 19 2019

When a general definition was made in a recent paper, it was slightly different from the previous definition. Please check the annotation on page 15 of the paper in 2019.

			Triangle begins:
      1,       1;
    2/3,     3/4,          1,       1;
   8/15,   41/64,      65/48,    11/8,     1/4,    1/4;
  16/35, 147/256, 13247/8640, 907/576, 109/216, 73/144, 1/36, 1/36;

Seiichi Manyama, Rows n = 0..15, flattened
Kazufumi Kimoto, Masato Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators, Kyushu Journal of Mathematics, Vol. 60 (2006) No. 2 p. 383-404 (see Table 2).
Kazufumi Kimoto, Masato Wakayama, Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals, arXiv:1905.01775 [math.PR], 2019. See p.22.

Cf. A056982 (k=2), A264542(n)/2 (k=3) (By the definition of A264542, Jtilde3(1)(1) = 1/2).

Cf. A001044, A326303 (numerator).

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def Jtilde(k, n)
  return 0 if k == 0
  return (2r ** n * f(n)) ** 2 / f(2 * n + 1) if k == 1
  if n == 0
    return 1 if k == 2
    return 0
  end
  if n == 1
    return 3r / 4 if k == 2
    return 1      if k == 3 || k == 4
    return 0
  end
  ((8r * n * n - 8 * n + 3) * Jtilde(k, n - 1) - 4 * (n - 1) ** 2 * Jtilde(k, n - 2) + 4 * Jtilde(k - 2, n - 1)) / (4 * n * n)
end
def A326748(n)
  (0..n).map{|i| (1..2 * i + 2).map{|j| Jtilde(j, i).denominator}}.flatten
end
p A326748(10)

4*n^2 * Jtilde_k(n) = (8*n^2 - 8*n + 3) * Jtilde_k(n-1) - 4*(n - 1)^2 * Jtilde_k(n-2) + 4 * Jtilde_{k - 2}(n-1).

Jtilde_n(2*n+1) = Jtilde_n(2*n+2) = 1/A001044(n). So T(n,2*n+1) = T(n,2*n+2) = A001044(n).

A327496 a(n) = a(n - 1) * 4^r where r = valuation(n, 2) if 4 divides n else r = (n mod 2) + 1; a(0) = 1. The denominators of A327495.

Original entry on oeis.org

1, 16, 64, 1024, 16384, 262144, 1048576, 16777216, 1073741824, 17179869184, 68719476736, 1099511627776, 17592186044416, 281474976710656, 1125899906842624, 18014398509481984, 4611686018427387904, 73786976294838206464, 295147905179352825856, 4722366482869645213696

Offset: 0

Peter Luschny, Sep 29 2019

Cf. A327491, A327492, A327493, A327494, A327495.

Cf. A056982.

A327496 := n -> denom(add(j!^2 / (2^j*iquo(j, 2)!)^4, j=0..n)):
seq(A327496(n), n=0..19);

a(n) = 1 << (4*n - 2*hammingweight(n>>1)); \\ Kevin Ryde, May 31 2022

@cached_function
def A327496(n):
    if n == 0: return 1
    r = valuation(n, 2) if 4.divides(n) else n % 2 + 1
    return 4^r * A327496(n-1)
print([A327496(n) for n in (0..19)])

a(n) = denominator(r(n)) where r(n) = Sum_{j=0..n} j!^2 / (2^j*floor(j/2))^4.

a(n) = 4^A327492(n). - Kevin Ryde, May 31 2022

A380949 a(n) = numerator(r(n)) where r(n) = (n/2)(Pi/2)^cos(Pi(n-1))*((n/2-1/2)!/(n/2)!)^2.

Original entry on oeis.org

0, 1, 1, 4, 9, 64, 75, 256, 1225, 16384, 19845, 65536, 160083, 1048576, 1288287, 4194304, 41409225, 1073741824, 1329696225, 4294967296, 10667118605, 68719476736, 85530896451, 274877906944, 1371086188563, 17592186044416, 21972535073125, 70368744177664, 176021737014375

Offset: 0

Peter Luschny, Feb 11 2025

			r(n) = 0, 1, 1/2, 4/3, 9/16, 64/45, 75/128, 256/175, 1225/2048, ...

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A380950 (denominator), A380910, A380909, A019267 (asymptotic coefficients).

Cf. A161736, A056982, A124399, A161737, A069955, A038534, A278145, A001901.

r := n -> (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2:
a := n -> numer(simplify(r(n))): seq(a(n), n = 0..28);
# Alternative:
r := n -> ifelse(n <= 1, n, (n - 1)/(n*r(n - 1))):

Join[{0}, Numerator[FoldList[(#2 - 1)/(#2*#) &, Range[30]]]] (* Paolo Xausa, Feb 14 2025 *)

Product_{k=1..n} a(k) = A380910(n) / A380909(n).
r(n) = (n - 1)/(n*r(n - 1)) for n > 1.
numerator(r(2*n)) = A161736(n).
numerator(r(2*n+1)) = A056982(n).
numerator(r(2*n+1))/4^n = A124399(n).
denominator(r(2*n-2)) = A161737(n).
denominator(r(2*n+1)) = A069955(n).
denominator(r(2*n+1))/(2*n+1) = A038534(n).
denominator(r(2*n+2))/2 = A278145(n).
denominator(r(2*n+2))/2^(2*n+1) = A001901(n).
r(n) ~ (2/Pi)^((-1)^n)*(1 - 1/(2*n) + 1/(8*n^2) + 1/(16*n^3) - 5/(128*n^4) - 23/(256*n^5) ...).

A056981 a(n) = A002596(n)^2.

Original entry on oeis.org

1, 1, 1, 1, 25, 49, 441, 1089, 184041, 511225, 5909761, 17631601, 863948449, 2704312009, 34493775625, 111759833025, 93990019574025, 312541206957225, 4201942893536025, 14258670483605625, 780804795682244025

Offset: 0

Eric W. Weisstein

Numerator of h^(2n) in the Kummer-Gauss series for the perimeter of an ellipse.

Eric Weisstein's World of Mathematics, Gauss-Kummer Series.
Eric Weisstein's World of Mathematics, Ellipse.

Cf. A002596, A056982.

Numerator[Table[Binomial[1/2, n]^2, {n, 50}]] (* Artur Jasinski, Aug 30 2016 *)

New description from N. J. A. Sloane, Feb 18 2004

A094083 Numerators of ratio of sides of n-th triple of rectangles of unit area sum around a triangle.

Original entry on oeis.org

1, 1, 1, 4, 9, 64, 25, 256, 1225, 16384, 3969, 65536, 53361, 1048576, 184041, 4194304, 41409225, 1073741824, 147744025, 4294967296, 2133423721, 68719476736, 7775536041, 274877906944, 457028729521, 17592186044416, 1690195005625

Offset: 1

Peter J. C. Moses, Apr 30 2004

Page 13 of the link shows the type of configuration. When n is odd, the numerators 1,1,9,25,1225,3969,.. are A038534 and (A001790)^2, and the denominators 1,4,64,256,16384,65536,.. are A056982, A038533/2, and (A046161)^2. When n is even, the numerators 1,4,64,256,16384,65536,.. are A056982, A038533/2, and (A046161)^2, and the denominators 3,27,675,3675,297675,1440747,.. are 3*(A001803)^2. The limit of a(n+1)/a(n) as n(odd) tends to infinity = Pi^2/12, A072691. The limit of a(n+2)/a(n) as n tends to infinity = 1. a(n), for large odd n, tends to 2/(Pi*n). a(n), for large even n, tends to Pi/(6*n). The expansion of 2*x*EllipticK(x)/Pi gives the odd fractions. The expansion of 1/3*x*HypergeometricPFQ({1,1,1},{3/2,3/2},x) gives the even fractions.

			a(5) = a(5-2)*((5-2)/(5-1))^2 = 1/4*(3/4)^2 = 9/64

Paul Yiu, EuclideanGeometry Notes

Cf. A005386, A092936, A094084.

a[n_]:=If[OddQ[n], ((n/2-1)!)^2/(Pi*((n/2-1/2)!)^2), Pi*((n/2-1)!)^2/(12*((n/2-1/2)!)^2)] a[n_]:=If[OddQ[n], (2^(1-n)*(n-2)!!^2)/((n-1)/2)!^2, (2^(n-2)*((n-2)/2)!^2)/(3*(n-1)!!^2)] a[n_]:=((12+Pi^2+E^(I*n*Pi)*(Pi^2-12))*((n/2-1)!)^2)/(24*Pi*((n/2-1/2)!)^2) (CoefficientList[Series[(I*x*(6+Sqrt[3]*Pi)-2*x*Sqrt[3]*Log[x+Sqrt[x^2-1]])/(6*Sqrt[x^2-1]), {x, 0, 20}], x])^2

a(n)=a(n-2)*((n-2)/(n-1))^2, a(1)=1, a(2)=1/3. a(n)=((n/2-1)!)^2/(Pi*((n/2-1/2)!)^2) for n odd. a(n)=(2^(1-n)*(n-2)!!^2)/((n-1)/2)!^2 for n odd. a(n)=Pi*((n/2-1)!)^2/(12*((n/2-1/2)!)^2) for n even. a(n)=(2^(n-2)*((n-2)/2)!^2)/(3*(n-1)!!^2) for n even.

A274657 Numerators of the coefficients of z^n/n! for the expansion of hypergeometric([1/2,1/2],[1];z).

Original entry on oeis.org

1, 1, 9, 75, 3675, 59535, 2401245, 57972915, 13043905875, 418854310875, 30241281245175, 1212400457192925, 213786613951685775, 10278202593831046875, 1070401384414690453125, 60013837619516978071875, 57673297952355815927071875, 3694483615889146090857721875

Offset: 0

Wolfdieter Lang, Jul 07 2016

The denominators are given in A123854.
The main entry is A038534 (with A056982) where comments and references are given.
The complete elliptic integral of the first kind K = K(k) is (Pi/2)*hypergeometric([1/2,1/2],[1];k^2). This is also the real quarter period K of elliptic functions.

			The first rationals r(n) are: 1, 1/4, 9/32, 75/128, 3675/2048, 59535/8192, 2401245/65536, 57972915/262144, 13043905875/8388608, 418854310875/33554432, 30241281245175/268435456, ...

Cf. A038534/A056982, A123854.

With[{n = 20}, Numerator[CoefficientList[Series[2 EllipticK[x]/Pi, {x, 0, n}], x] Range[0, n]!]] (* Jan Mangaldan, Jan 04 2017 *)
Numerator[Table[Gamma[n + 1/2]^2/(Pi Gamma[n + 1]), {n, 0, 20}]] (* Li Han, Feb 05 2021 *)

a(n) = numerator(r(n)) with the rationals (in lowest terms) r(n) = (risefac(1/2,n)^2)/n! = ((2*n)!^2)/((n!^3)*2^(4*n)), with the rising factorial risefac (Pochhammer symbol).

E.g.f. for r(n) is hypergeometric([1/2,1/2],[1];z).

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 2cos(v) with v = u/((2/Pi)K(k)).

Original entry on oeis.org

1, -4, 1, 4, -5, 1, 0, 12, -7, 1, 4, -21, 25, -9, 1, -8, 30, -63, 42, -11, 1, 0, -44, 131, -138, 63, -13, 1, 0, 72, -246, 365, -253, 88, -15, 1, 4, -85, 425, -837, 808, -416, 117, -17, 1, -4, 85, -685, 1734, -2200, 1552, -635, 150, -19, 1, 8, -134, 1053, -3319, 5326, -4888, 2705, -918, 187, -21, 1

Offset: 0

Wolfdieter Lang, Aug 10 2016

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
  cn(u, k) = (theta_4(0, q)/theta_2(0, q)) * (theta_2(v, q)/theta_4(v, q)).
  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)).
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).
The product representation involves directly (2*cos(v))^2 powers in the q expansion:
  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).
For another version of this cn expansion see A274661.
For the sn(u, k)/sin(v) analog see A274662.
This entry is inspired by a paper of Bradley Klee giving an approximation to the phase space solution of the plane pendulum (see A273506).
See also the W. Lang link, equations (59) and (60).

			The triangle T(n, m) begins:
n\m 0   1    2    3    4    5    6   7   8 9
0:   1
1:  -4   1
2:   4  -5    1
3:   0  12   -7    1
4:   4 -21   25   -9     1
5:  -8  30  -63   42   -11    1
6:   0 -44  131 -138    63  -13    1
7:   0  72 -246  365  -253   88  -15   1
8:   4 -85  425 -837   808 -416  117 -17   1
9:  -4  85 -685 1734 -2200 1552 -635 150 -19 1
...
Row n=10: 8 -134 1053 -3319 5326 -4888 2705 -918 187 -21 1.
...
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.
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).

F. Tricomi, Elliptische Funktionen (German translation by M. Krafft of: Funzioni ellittiche), Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1948.

Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum

Cf. A005797, A002103, A038534/A056982, A053120, A274661, A274662.

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.

cp's OEIS Frontend

A124399 a(n) = 4^(n - bitcount(n)) where bitcount(n) = A000120(n).

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A274662 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function sn(u, k) divided by sin(v) in terms of the Jacobi nome q and even powers of 2*cos(v), with v = u/((2/Pi)*K(k)).

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A326748 Triangular array, read by rows: T(n,k) = denominator of Jtilde_k(n), 1 <= k <= 2*n+2.

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A327496 a(n) = a(n - 1) * 4^r where r = valuation(n, 2) if 4 divides n else r = (n mod 2) + 1; a(0) = 1. The denominators of A327495.

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A380949 a(n) = numerator(r(n)) where r(n) = (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2.

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A056981 a(n) = A002596(n)^2.

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A094083 Numerators of ratio of sides of n-th triple of rectangles of unit area sum around a triangle.

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A274662 Triangle T(n, m) appearing in the expansion of Jacobi's elliptic function sn(u, k) divided by sin(v) in terms of the Jacobi nome q and even powers of 2cos(v), with v = u/((2/Pi)K(k)).

A380949 a(n) = numerator(r(n)) where r(n) = (n/2)(Pi/2)^cos(Pi(n-1))*((n/2-1/2)!/(n/2)!)^2.

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 2cos(v) with v = u/((2/Pi)K(k)).