A002103 -id:A002103 - cp's OEIS frontend

A079006 Expansion of q^(-1/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q.

1, -2, 5, -10, 18, -32, 55, -90, 144, -226, 346, -522, 777, -1138, 1648, -2362, 3348, -4704, 6554, -9056, 12425, -16932, 22922, -30848, 41282, -54946, 72768, -95914, 125842, -164402, 213901, -277204, 357904, -460448, 590330, -754368, 960948, -1220370

Offset: 0

Michael Somos, Dec 22 2002

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The Lagrange series reversion of Sum_{n >= 1} a(n-1)*x^n is Sum_{n >= 1} A002103(n-1)*x^n. See the example in A002103. - Wolfdieter Lang, Jul 09 2016

			G.f. A(x) = 1 - 2*x + 5*x^2 - 10*x^3 + 18*x^4 - 32*x^5 + 55*x^6 - 90*x^7 + 144*x^8 + ...
G.f. B(q) = q * A(q^4) = q - 2*q^5 + 5*q^9 - 10*q^13 + 18*q^17 - 32*q^21 + 55*q^25 - 90*q^29 + ...

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012, eq. (8), p. 11 with eq. (2), p. 10.
Nicco, Conjectured identity of the product of two theta functions, Math Stackexchange, Sep 09 2015.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000700, A001936, A002103, A008441, A029839, A046897, A083365, A127391, A127392, A189925.

a[ n_] := SeriesCoefficient[ Product[(1 + x^(k + 1)) / (1 + x^k), {k, 1, n, 2}]^2, {x, 0, n}]; (* Michael Somos, Jul 08 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ[ q]}, SeriesCoefficient[ (m / 16 / q)^(1/4), {q, 0, n}]]; (* Michael Somos, Jul 08 2011 *)
QP = QPochhammer; s = (QP[q]*(QP[q^4]^2/QP[q^2]^3))^2 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 23 2015 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(2*k))^4 / (1+x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^4]^2 / QPochhammer[ -x]^2, {x, 0, n}]; (* Michael Somos, Apr 19 2017 *)

{a(n) = my(N, A); if( n<0, 0, N = (sqrtint(16*n + 1) + 1)\2; A = contfracpnqn( matrix(2, N, i, j, if( i==1, if( j<2, 1 + O(x^(N^2 + N)), (x^(j-1) + x^(3*j - 3))^2), 1 - x^(4*j - 2)))); polcoeff( A[2,1] / A[1,1], 4*n))}; /* Michael Somos, Sep 01 2005 */

{a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m = 1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A / (1 + 4 * x*A^2))); polcoeff(A, n))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^2, n))};

a(n) = (2/n)*Sum_{k=1..n} (-1)^k*A046897(k)*a(n-k). - Vladeta Jovovic, Dec 24 2002
Expansion of q^(-1/4) * (1/2) * k^(1/2) in powers of q, where k^2 is the parameter and q the Jacobi nome of elliptic functions.
Expansion of (1/(2*q)) * (1 - sqrt(k')) / (1 + sqrt(k')) in powers of q^4, where k'^2 is the complementary parameter and q the Jacobi nome of elliptic functions. See the Fricke reference.
Expansion of psi(x^2) / phi(x) = psi(x)^2 / phi(x)^2 = psi(x^2)^2 / psi(x)^2 = psi(-x)^2 / phi(-x^2)^2 = chi(-x)^2 / chi(-x^2)^4 = 1 / (chi(x)^2 * chi(-x^2)^2) = 1 / (chi(x)^4 * chi(-x)^2) = f(-x^4)^2 / f(x)^2 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [-2, 4, -2, 0, ...].
G.f. A(x) satisfies A(x)^2 = A(x^2) / (1 + 4 * x * A(x^2)^2). - Michael Somos, Mar 19 2004
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 * (1 + 4 * v^2) - v. - Michael Somos, Jul 09 2005
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u3 * (u6 + u2)^2 - u2*u6. - Michael Somos, Jul 09 2005
G.f.: (Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k-1)))^2 = (Product_{k>0} (1 - x^(4*k)) / (1 - (-x)^k))^2.
Expansion of continued fraction 1 / (1 - x^2 + (x^1 + x^3)^2 / (1 - x^6 + (x^2 + x^6)^2 / (1 - x^10 + (x^3 + x^9)^2 / ...))) in powers of x^4. - Michael Somos, Sep 01 2005
Given g.f. A(x), then B(q) = 2 * q * A(q^4) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 - u^4) * (1 - v^4) - (1 - u*v)^4 . - Michael Somos, Jan 01 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A189925.
Convolution inverse of A029839. Convolution square of A083365. a(n) = (-1)^n * A001936(n).
G.f.: 1/Q(0), where Q(k)= 1 - x^(k+1/2) + (x^((k+1)/4) + x^((3*k+3)/4))^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 02 2013
a(n) ~ (-1)^n * exp(Pi*sqrt(n)) / (2^(7/2)*n^(3/4)). - Vaclav Kotesovec, Jul 04 2016
Given g.f. A(x), and B(x) is the g.f. for A008441, then A(x) = B(x^2) / B(x) and A(x) * A(x^2) * A(x^4) * ... = 1 / B(x). - Michael Somos, Apr 20 2017
Expansion of continued fraction 1 / (1 - x^1 + x^1*(1 + x^1)^2 / (1 - x^3 + x^2*(1 + x^2)^2 / (1 - x^5 + x^3*(1 + x^3)^2 / ...))) in powers of x^2. - Michael Somos, Apr 20 2017
a(n) = A208933(4*n+1) - A215348(4*n+1) (conjectured). - Thomas Baruchel, May 14 2018
A(x^4) = (1/(m*x)) * ( chi(x)^m - chi(-x)^m ) / ( chi(x)^m + chi(-x)^m ) at m = 2, where chi(x)  = Product_{i >= 0} (1 + x^(2*i+1)) is the g.f. of A000700. The formula gives generating functions related to A092869 when m = 1 and A001938 (also A093160) when m = 4. - Peter Bala, Sep 23 2023

A005797 Expansion of Jacobi nome q in terms of parameter m/16.

Original entry on oeis.org

0, 1, 8, 84, 992, 12514, 164688, 2232200, 30920128, 435506703, 6215660600, 89668182220, 1305109502496, 19138260194422, 282441672732656, 4191287776164504, 62496081197436736, 935823746406530603, 14065763582458332888, 212122153814497767004, 3208590886304243284640

Offset: 0

N. J. A. Sloane

For a faster convergent series see A002103, where k' = sqrt(1 - k^2). - Wolfdieter Lang, Jul 14 2016

The Ansatz technique of A308835, A308836, and A308837 also works to produce the coefficients of this sequence from the ODE: T-d/dx(4*(1-x)*x*dT/dx)=0. - Bradley Klee, Jul 03 2019

			G.f. = x + 8*x^2 + 84*x^3 + 992*x^4 + 12514*x^5 + 164688*x^6 + 2232200*x^7 + ...
Given g.f. A(x),  then q = exp(-Pi sqrt(6)) = A( m/16 ) where m = ((2-sqrt(3))*(sqrt(3)-sqrt(2)))^2. - _Michael Somos_, Oct 30 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054.
C. L. Mallows, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..700
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 591.
Index entries for reversions of series

Reversion of A005798. Cf. A002639. Other nomes: A308835, A308836, A308837.

a:= n-> coeff(series(EllipticNome(4*sqrt(x)), x, n+1), x, n):
seq(a(n), n=0..17);  # Thomas Richard, Aug 03 2022

a[ n_] := SeriesCoefficient[ EllipticNomeQ[ 16 x], {x, 0 ,n}] (* Michael Somos, Jul 11 2011 *)

{a(n) = if( n < 1, 0, polcoeff( serreverse( x * prod(k=1, n-1, (1 + x^k)^(-1)^k, 1 + x * O(x^n))^8), n))} /* Michael Somos, Jul 19 2002 */

{a(n) = my(A, m); if( n < 1, 0, m=1; A = x + O(x^2); while( m < n, m*=2; A = sqrt( subst(A, x, x^2)); A /= (1 + 4*A)^2); polcoeff( serreverse(A), n))} /* Michael Somos, Mar 18 2003 */

G.f.: q = q(m) = Sum_{n>=0} a(n) * (m/16)^n.

G.f.: exp( -Pi * agm(1, sqrt(1 - 16 * x) ) / agm(1, sqrt( 16*x ) ) ).

A002639 Numerators of expansion of Jacobi nome q in parameter m.

Original entry on oeis.org

0, 1, 1, 21, 31, 6257, 10293, 279025, 483127, 435506703, 776957575, 22417045555, 40784671953, 9569130097211, 17652604545791, 523910972020563, 976501268709949, 935823746406530603, 1758220447807291611

Offset: 0

N. J. A. Sloane

			q = 1/16*m + 1/32*m^2 + 21/1024*m^3 + 31/2048*m^4 + 6257/524288*m^5 + ...

Guide to Tables, Math. Tables Other Aids Computation, 3 (1948), Section III, p. 234.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Charles Hermite, Oeuvres.
Charles Hermite, Sur Quelques Développements En Série de la Théorie Des Fonctions Elliptiques , Oeuvres. Vol. 4, Gauthier-Villars, Paris, 1917, p. 477.
Charles Hermite, Annotated scan of a page from the Oeuvres, together with a page from Math. Tables Aids Comp., Vol. 3, 1948 that refers to it.
J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques (Vol. 4), Gauthier-Villars, Paris, 1902, p. 141.
Eric Weisstein's World of Mathematics, Nome
Robert M. Ziff, On Cardy's formula for the critical crossing probability in 2d percolation, J. Phys. A. 28, 1249-1255 (1995).

Cf. A119349 (denominators), A002103 (where there are further references), A005797.

A002639 := proc(n::integer)
    local z;
    # coeftayl(EllipticNome(z),z=0,n) ; # very slow
    taylor(EllipticNome(z),z=0,2*n+1) ;
    convert(%,polynom) ;
    coeff(%,z,2*n) ;
    numer(%) ;
end proc:
seq(A002639(n),n=0..10) ; # R. J. Mathar, Mar 26 2025

Numerator[ CoefficientList[ Series[ EllipticNomeQ[m], {m, 0, 18}], m]] (* Jean-François Alcover, Sep 21 2011 *)

{a(n) = if( n<1, 0, numerator( polcoeff( serreverse( x * prod(k=1, n, (1 + x^k)^(-1)^k, 1 +x * O(x^n))^8), n) / 4^n))}

Edited by Michael Somos, Aug 09 2002

A274659 Triangle entry T(n, m) gives the m-th contribution T(n, m)sin((2m+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/(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

Offset: 0

Wolfdieter Lang, Jul 18 2016

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).
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 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.
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.
For the cn expansion see A274661.
See also the W. Lang link, equations (34) and (35).

			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
...
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.
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.

J. V. Armitage and W. F. Eberlein, Elliptic Functions, London Mathematical Society, Student Texts 67, Cambridge University Press, 2006.
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, 16.23.1.
Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum

Cf. A002103, A038534/A056982, A099774, A274621, A274658, A274661.

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.

A293725 Numbers k such that c(k,0) = c(k,1), where c(k,d) = number of d's in the first k digits of the base-2 expansion of sqrt(2).

Original entry on oeis.org

2, 10, 20, 24, 28, 32, 318, 328, 330, 334, 336, 608, 622, 636, 638, 674, 676, 678, 680, 682, 826, 828, 832, 836, 838, 842, 844, 846, 848, 850, 852, 856, 858, 876, 880, 884, 886, 898, 906, 908, 918, 920, 928, 930, 942, 944, 946, 948, 950, 962, 964, 966, 968

Offset: 1

Clark Kimberling, Oct 16 2017

This sequence together with A293727 and A293728 partition the positive integers.

			In base 2, sqrt(2) = 1.0110101000001001111001..., so that initial segments 1.0; 1.011010100..., of lengths 2,10,... have the same number of 0's and 1's.

Cf. A004539, A002103, A293726, A293727, A293728.

z = 300; u = N[Sqrt[2], z]; d = RealDigits[u, 2][[1]];
t[n_] := Take[d, n]; c[0, n_] := Count[t[n], 0]; c[1, n_] := Count[t[n], 1];
Table[{n, c[0, n], c[1, n]}, {n, 1, 100}]
u = Select[-1 + Range[z], c[0, #] == c[1, #] &]  (* A293725 *)
u/2  (* A293726 *)
Select[-1 + Range[z], c[0, #] < c[1, #] &]  (* A293727 *)
Select[-1 + Range[z], c[0, #] > c[1, #] &]  (* A293728 *)

A293726 a(n) = (1/2)*A293725(n).

Original entry on oeis.org

1, 5, 10, 12, 14, 16, 159, 164, 165, 167, 168, 304, 311, 318, 319, 337, 338, 339, 340, 341, 413, 414, 416, 418, 419, 421, 422, 423, 424, 425, 426, 428, 429, 438, 440, 442, 443, 449, 453, 454, 459, 460, 464, 465, 471, 472, 473, 474, 475, 481, 482, 483, 484

Offset: 1

Clark Kimberling, Oct 18 2017

Cf. A004539, A002103, A293726, A293727, A293728.

z = 300; u = N[Sqrt[2], z]; d = RealDigits[u, 2][[1]];
t[n_] := Take[d, n]; c[0, n_] := Count[t[n], 0]; c[1, n_] := Count[t[n], 1];
Table[{n, c[0, n], c[1, n]}, {n, 1, 100}]
u = Select[Range[z], c[0, #] == c[1, #] &]  (* A293725 *)
u/2  (* A293726 *)
Select[Range[z], c[0, #] < c[1, #] &]  (* A293727 *)
Select[Range[z], c[0, #] > c[1, #] &]  (* A293728 *)

A293727 Numbers k such that c(k,0) < c(k,1), where c(k,d) = number of d's in the first k digits of the base-2 expansion of sqrt(2).

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 9, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91

Offset: 1

Clark Kimberling, Oct 18 2017

This sequence together with A293725 and A293728 partition the nonnegative integers.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A004539, A002103, A293726, A293727, A293728.

z = 300; u = N[Sqrt[2], z]; d = RealDigits[u, 2][[1]];
t[n_] := Take[d, n]; c[0, n_] := Count[t[n], 0]; c[1, n_] := Count[t[n], 1];
Table[{n, c[0, n], c[1, n]}, {n, 1, 100}]
u = Select[Range[z], c[0, #] == c[1, #] &]  (* A293725 *)
u/2  (* A293726 *)
Select[Range[z], c[0, #] < c[1, #] &]  (* A293727 *)
Select[Range[z], c[0, #] > c[1, #] &]  (* A293728 *)

A293728 Numbers k such that c(k,0) > c(k,1), where c(k,d) = number of d's in the first k digits of base-2 expansion of sqrt(2).

Original entry on oeis.org

11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 335, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 675, 677, 683, 684, 685, 686, 687, 688, 689, 690

Offset: 1

Clark Kimberling, Oct 18 2017

This sequence together with A293725 and A293727 partition the nonnegative integers.

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A004539, A002103, A293726, A293727, A293728.

z = 300; u = N[Sqrt[2], z]; d = RealDigits[u, 2][[1]];
t[n_] := Take[d, n]; c[0, n_] := Count[t[n], 0]; c[1, n_] := Count[t[n], 1];
Table[{n, c[0, n], c[1, n]}, {n, 1, 100}]
u = Select[Range[z], c[0, #] == c[1, #] &]  (* A293725 *)
u/2  (* A293726 *)
Select[Range[z], c[0, #] < c[1, #] &]  (* A293727 *)
Select[Range[z], c[0, #] > c[1, #] &]  (* A293728 *)

A227503 q = x * exp( 8 * (Sum_{k>0} a(k) * x^k / k)) where x = m/16, q is the elliptic nome and m = k^2 is the parameter.

Original entry on oeis.org

1, 13, 184, 2701, 40456, 613720, 9391936, 144644749, 2238445480, 34772271208, 541801226176, 8463116730712, 132472258939840, 2077232829015616, 32621327116946944, 512963507737401997, 8075477240446327528, 127258797512376887176, 2007225253307641799872

Offset: 1

Michael Somos, Jul 13 2013

nonn

The Fricke reference has equation Pi i omega / 4 = log (sqrt(k) / 2) + 2 (sqrt(k) / 2)^4 + 13 (sqrt(k) / 2)^8 + 368/3 (sqrt(k) / 2)^12 + 2701/2 (sqrt(k) / 2)^16 + ... .

This can be written (with Pi i omega / 4 = log(q)/4) as (log(q) - log(k^2/16)) / (8*k^2/16) = Sum_{n >= 0} (a(n+1)/(n+1))*(k^2/16)^n. See also the Kneser reference, p. 216. Note that the rational coefficients a(n+1)/(n+1) are not reduced to lowest terms. For the reduced rational coefficients see A274345 / A274346. - Wolfdieter Lang, Jun 30 2016

			G.f. = x + 13*x^2 + 184*x^3 + 2701*x^4 + 40456*x^5 + 613720*x^6 + 9391936*x^7 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 1..800
A. Kneser, Neue Untersuchung einer Reihe aus der Theorie der elliptischen Funktionen, J. reine u. angew. Math. 157, 1927, 209 - 218.
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012., p. 2, eq. (5).

Cf. A002103, A274345, A274346.

a[ n_] := If[ n < 0, 0, n SeriesCoefficient[ Log[ EllipticNomeQ[ 16 x] / x] / 8, {x, 0, n}]];

{a(n) = local(A); if( n<1, 0, A = x * O(x^n); n * polcoeff( log( serreverse( x * (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^8 ) / x) / 8, n))};

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.

cp's OEIS Frontend

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