A108494 -id:A108494 - cp's OEIS frontend

A080054 G.f.: Product_{n >= 0} (1+x^(2n+1))/(1-x^(2n+1)).

1, 2, 2, 4, 6, 8, 12, 16, 22, 30, 40, 52, 68, 88, 112, 144, 182, 228, 286, 356, 440, 544, 668, 816, 996, 1210, 1464, 1768, 2128, 2552, 3056, 3648, 4342, 5160, 6116, 7232, 8538, 10056, 11820, 13872, 16248, 18996, 22176, 25844, 30068, 34936, 40528

Offset: 0

Michael Somos, Jan 26 2003

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
G.f. for pairs of partitions of type R.
G.f. for the number of partitions of 2n in which all odd parts occur with multiplicity 2 and the even parts occur with multiplicity 1. Also g.f. for the number of partitions of 2n free of multiples of 4. All odd parts occur with even multiplicities. The even parts occur with multiplicity 1. - Noureddine Chair, Feb 10 2005
This is also the number of overpartitions of an integer into odd parts. - James Sellers, Feb 18 2008
The Higher Algebra reference on page 517 has an unnumbered example between 251 and 252: "If u^6-v^6+5u^2v^2(u^2-v^2)+4uv(1-u^4v^4)=0, prove that (u^2-v^2)^6=16u^2v^2(1-u^8)(1-v^8). [PEMB. COLL. CAMB.]". It turns out that this is two forms of the modular equation of degree 5. - Michael Somos, May 12 2011
Convolution of A000009 and A000700. - Vaclav Kotesovec, Aug 23 2015
Let F(x) = Product_{n >= 0} (1 + x^(2*n+1))/(1 - x^(2*n+1)). Let m be a nonzero integer. The simple continued fractions expansions of the real numbers F(1/m) may be predictable - given a positive integer n, the sequence of the n-th partial denominators of the continued fraction expansion of F(1/m) may be polynomial or quasi-polynomial in m for sufficiently large m. An example is given below. - Peter Bala, Nov 03 2019

			G.f. = 1 + 2*q + 2*q^2 + 4*q^3 + 6*q^4 + 8*q^5 + 12*q^6 + 16*q^7 + 22*q^8 + 30*q^9 + ...
From _Peter Bala_, Nov 03 2019: (Start)
F(x) := Product_{n >= 0} (1 + x^(2*n+1))/(1 - x^(2*n+1)).
Simple continued fraction expansions of F(1/(2*m)):
  m=2 [1; 1, 2, 1, 1, 1, 1, 2, 1, 2,   33, 1, 3,  7, 4,  33, 1, 8,  4,    2, 1,...]
  m=3 [1; 2, 2, 2, 1, 1, 2, 2, 2, 2,  110, 1, 2, 46, 3, 110, 1, 3, 12,    1, 7,...]
  m=4 [1; 3, 2, 3, 1, 1, 3, 2, 3, 2,  259, 1, 1,  1, 2,  15, 2, 1,  2,  259, 1,...]
  m=5 [1; 4, 2, 4, 1, 1, 4, 2, 4, 2,  504, 1, 1,  1, 1,  78, 1, 1,  2,  504, 1,...]
  m=6 [1; 5, 2, 5, 1, 1, 5, 2, 5, 2,  869, 1, 1,  2, 2,  23, 2, 2,  2,  869, 1,...]
  m=7 [1; 6, 2, 6, 1, 1, 6, 2, 6, 2, 1378, 1, 1,  2, 1, 110, 1, 2,  2, 1378, 1,...]
  m=8 [1; 7, 2, 7, 1, 1, 7, 2, 7, 2, 2055, 1, 1,  3, 2,  31, 2, 3,  2, 2055, 1,...]
  m=9 [1; 8, 2, 8, 1, 1, 8, 2, 8, 2, 2924, 1, 1,  3, 1, 142, 1, 3,  2, 2924, 1,...]
The sequence of the 10th partial denominators [33,110,259,504,...], starting at m = 2, appears to be given by the polynomial 4*m^3 + m - 1.
The sequence of the 15th partial denominators [15,78,23,110,31,142,...], starting at m = 4, appears to be quasi-polynomial in m, with constituent polynomials 4*m - 1 and 16*m - 2. (End)

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.
A. Cayley, An Elementary Treatise on Elliptic Functions, 2nd ed., G. Bell and Sons, 1895, p. 245, Art. 333.
J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. VI
J. W. L. Glaisher, On Some Continued Fractions, Messenger of Mathematics, 7 (1878), pp. 67-68, see p. 68
H. S. Hall and S. R. Knight, Higher Algebra, Macmillan, 1957, p. 517.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eddy Ardonne, Rinat Kedem, and Michael Stone, Filling the Bose sea: symmetric quantum Hall edge states and affine characters, arXiv:cond-mat/0409369 [cond-mat.mes-hall], 2004; Journal of Physics A: Mathematical and General 38.3 (2005): 617. - From _N. J. A. Sloane_, Apr 24 2014
C. Bessenrodt, On pairs of partitions with steadily decreasing parts, J. Combin. Theory, A 99 (2002), 162-174. MR1911463 (2003c:11133).
A. Cayley, A memoir on the transformation of elliptic functions, Philosophical Transactions of the Royal Society of London, 164 (1874), pp. 397-456, see pages 424 and 430.
Shi-Chao Chen, On the number of overpartitions into odd parts, Discrete Math. 325 (2014), 32--37. MR3181230. But beware of typos in the g.f. on page 32. - _N. J. A. Sloane_, Apr 24 2014
B. Hemanthkumar and S. Chandankumar, New congruences modulo small powers of 2 for overpartitions into odd parts, Matematički Vesnik (2020).
M. D. Hirschhorn and J. A. Sellers, Arithmetic Properties of Overpartitions into Odd Parts, Annals of Combinatorics 10, no. 3 (2006), 353-367.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 11.
Mircea Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69. See q-bar(n) on p. 66.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See q-bar(n).
Vladimir Reshetnikov, A conjecture about algebraic values of (-q;-q)_oo/(q;q)_oo, Math Overflow Posting, Nov 23 2016, with proof supplied by Noam D. Elkies.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Elliptic Lambda Function

Cf. A000384, A007096, A029838, A080015, A098151, A103258, A103260, A108494, A215594, A261610, A261611.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-2) +add(2*b(n-i*j, i-2), j=1..n/i)))
    end:
a:= n-> b(n, n-1+irem(n, 2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 10 2014
# alternative program using expansion of f(x, x^3) / f(-x, -x^3):
with(gfun): series( add(x^(n*(2*n-1)), n = -8..8)/add((-1)^n*x^(n*(2*n-1)), n = -8..8), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m )^(-1/8), {q, 0, n}]]; (* Michael Somos, Aug 03 2011 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q])^(1/2), {q, 0, n}]; (* Michael Somos, Aug 03 2011 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q] / QPochhammer[ q], {q, 0, n}]; (* Michael Somos, May 10 2014 *)
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-1}, {}, q^2, q], {q, 0, n}]; (* Michael Somos, May 10 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 2] + Sum[2*b[n - i*j, i - 2], {j, 1, n/i}]]];
a[n_] := b[n, n - 1 + Mod[n, 2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 05 2017, after Alois P. Heinz *)

{a(n) = my(A, m); if( n<0, 0, m=1; A = 1 + 2*x + O(x^2); while( m

a(n)=polcoeff(exp(2*sum(k=0,n\2,sigma(2*k+1)/(2*k+1)*x^(2*k+1))),n) /* Paul D. Hanna */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)), n))}; /* Michael Somos, Jul 07 2005 */

Expansion of f(q) / f(-q) in powers of q where f() is a Ramanujan theta function.
Expansion of (1 - k^2)^(-1/8) = k'^(-1/4) in powers of the nome q = exp(-Pi K'/K).
Expansion of eta(q^2)^3 / (eta(q^4) * eta(q)^2) in powers of q.
Euler transform of period 4 sequence [ 2, -1, 2, 0, ...].
(theta_3(q) / theta_4(q))^(1/2) = (phi(q) / phi(-q))^(1/2) = chi(q) / chi(-q) = psi(q) / psi(-q) = f(q) / f(-q) where phi{}, chi(), psi(), f() are Ramanujan theta functions.
G.f.: A(x) = exp( 2*sum_{n>=0} sigma(2*n+1)/(2*n+1)*x^(2*n+1) ). - Paul D. Hanna, Mar 01 2004
G.f. satisfies: A(-x) = 1/A(x), (A(x)+A(-x))/2 = A(x^2)*A(x^4)^2, A(x) = sqrt((A(x^2)^4+1)/2) + sqrt((A(x^2)^4-1)/2). - Paul D. Hanna, Mar 27 2004
Another g.f.: 1/product_{ k>= 1 } (1+x^(2*k))*(1-x^(2*k-1))^2. - Vladeta Jovovic, Mar 29 2004
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v^3) * (v + 2*u^3) - u * (u^3 - v). - Michael Somos, Aug 03 2011
G.f. A(x) satisfies 0 = f(A(x), A(x^5)) where f(u, v) = (u^2 - v^2)^6 - 16 * u^2 * v^2 * (1 - u^8) * (1 - v^8). - Michael Somos, May 12 2011
G.f. A(x) satisfies 0 = f(A(x), A(x^7)) where f(u, v) = (1 - u^8) * (1 - v^8) - (1 - u*v)^8. - Michael Somos, Jan 01 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A029838. - Michael Somos, Aug 03 2011
G.f.: (theta_3/theta_4)^(1/2) = ((Sum_{k in Z} x^(k^2))/(Sum_{k in Z} (-x)^(k^2)))^(1/2) = Product_{k>0} (1 - x^(4k-2))/((1 - x^(4k-1))(1 - x^(4k-3)))^2.
G.f.: Product_{ k >= 1 } (1 + x^(2*k-1))*(1 + x^k) = product_{ k >= 1 } (1 + x^(2*k-1))/(1 - x^(2*k-1)).
G.f.: 1 + 2*x / (1 - x) + 2*x^3 * (1 + x) / ((1 - x)*(1 - x^2)) + 2*x^6 * (1 + x)*(1 + x^2) / ((1 - x)*(1 - x^2)*(1 - x^3)) + ... [Glaisher 1876] - Michael Somos, Jun 20 2012
G.f.: 1 / (1 - 2*x / (1 + x - (x^2 - x^4) / (1 + x^3 - (x^3 - x^7) / (1 + x^5 - (x^4 - x^10) / (1 + x^7 - ...))))) [Glaisher 1878] - Michael Somos, Jun 24 2012
a(n) = (-1)^floor(n/2) * A080015(n) = (-1)^n * A108494(n). Convolution inverse is A108494. Convolution square is A007096.
Empirical : Sum_{n>=0} exp(-Pi)^n * a(n) = 2^(1/8). - Simon Plouffe, Feb 20 2011
Empirical : Sum_{n>=0} (-exp(-Pi))^n * a(n) = 1/2^(1/8). - Simon Plouffe, Feb 20 2011
a(n) ~ Pi * BesselI(1, Pi*sqrt(n/2)) / (4*sqrt(n)) ~ exp(Pi*sqrt(n/2)) / (2^(9/4) * n^(3/4)) * (1 - 3/(4*Pi*(sqrt(2*n))) - 15/(64*Pi^2*n)). - Vaclav Kotesovec, Aug 23 2015, extended Jan 09 2017
Simon Plouffe's empirical observations are true. Furthermore, for every positive rational p, Sum_{n>=0} exp(-Pi*sqrt(p))^n * a(n) = 1/(Sum_{n>=0} (-exp(-Pi*sqrt(p)))^n * a(n)) is an algebraic number (see the MathOverflow link). - Vladimir Reshetnikov, Nov 23 2016
G.f.: f(x,x^3)/f(-x,-x^3) = ( Sum_{n = -oo..oo} x^(n*(2*n-1)) )/( Sum_{n = -oo..oo} (-1)^n*x^(n*(2*n-1)) ), where f(a,b) = Sum_{n = -oo..oo} a^(n*(n+1)/2)*b^(n*(n-1)/2) is Ramanujan's 2-variable theta function. - Peter Bala, Feb 05 2021
G.f. A(q) = (-lambda(-q)/lambda(q))^(1/8), where lambda(q) = 16*q - 128*q^2 + 704*q^3 - 3072*q^4 + ... is the elliptic modular function in powers of the nome q = exp(i*Pi*t), the g.f. of A115977; lambda(q) = k(q)^2, where k(q) = (theta_2(q) / theta_3(q))^2 is the elliptic modulus. - Peter Bala, Sep 26 2023
Recurrence: a(n) = c(n) + Sum_{k = 1..floor((-1 + sqrt(1 + 8*n))/2)} (-1)^(1 + k*(k+1)/2) * a(n - k*(k+1)/2), where c(n) = 1 if n is a triangular number, otherwise c(n) = 0. See A010054. - Peter Bala, Jun 08 2025

Definition simplified by N. J. A. Sloane, Apr 24 2014

A083365 Expansion of psi(x) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 2, -3, 4, -6, 9, -12, 16, -22, 29, -38, 50, -64, 82, -105, 132, -166, 208, -258, 320, -395, 484, -592, 722, -876, 1060, -1280, 1539, -1846, 2210, -2636, 3138, -3728, 4416, -5222, 6163, -7256, 8528, -10006, 11716, -13696, 15986, -18624, 21666, -25169, 29190, -33808, 39104

Offset: 0

Michael Somos, Apr 24 2003

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square is A079006.
Convolution inverse is A029838.

			G.f. = 1 - x + 2*x^2 - 3*x^3 + 4*x^4 - 6*x^5 + 9*x^6 - 12*x^7 + 16*x^8 - 22*x^9 + ...
G.f. = q - q^9 + 2*q^17 - 3*q^25 + 4*q^33 - 6*q^41 + 9*q^49 - 12*q^57 + 16*q^65 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 221 Entry 1(i).
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.
H. T. Davis, Introduction to nonlinear differential and integral equations, Dover Publications, Inc., New York 1962, p. 170 MR0181773 (31 #6000)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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]
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162; see Eqs. (9.1),(9.3).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions, q-Pochhammer Symbol

Cf. A001935, A029838, A108494.

(psi(x) / phi(x))^b: this sequence (b=1), A079006 (b=2), A187053 (b=3), A001938 (b=4), A195861 (b=5), A320049 (b=6), A320050 (b=7).

phi[x_] := EllipticTheta[3, 0, x]; psi[x_] := (1/2)*x^(-1/8)*EllipticTheta[2, 0, x^(1/2)]; s = Series[ psi[x]/phi[x], {x, 0, 100}]; A083365 = CoefficientList[s, x] (* Jean-François Alcover, Feb 18 2015 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k))^2/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 04 2016 *)
(QPochhammer[-x^2, x^2, -1/2] + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^2] / QPochhammer[ -x, x^2], {x, 0, n}]; (* Michael Somos, Oct 10 2019~ *)

{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(sqrt(A), n))};

{a(n) = my(A); if( n<0, 0, A = contfracpnqn( matrix(2, (sqrtint(8*n + 1) + 1)\2, i, j, if( i==1, x^(j-1), 1 + if( j>1, x^(j-1))))); polcoeff(A[2,1] / A[1,1] + x * O(x^n), 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, n))};

Expansion of f(-x^4) / f(x) = psi(x) / phi(x) = psi(x^2) / psi(x) = psi(-x) / phi(-x^2) = 1 / (chi(x) * chi(-x^2)) = 1 / (chi^2(x) * chi(-x)) = chi(-x) / chi^2(-x^2) = (psi(x^2) / phi(x))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of k^(1/4) / (2^(1/2) * q^(1/8)) in powers of q where k is elliptic modulus and q is the nome.
Expansion of q^(-1/8) * eta(q) * eta(q^4)^2 / eta(q^2)^3 in powers of q.
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - u^4 * (1 + 4*v^4).
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = v^4 - u^4 + u*v + 4*(u*v)^3.
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = w - u^2*v*(1 + 2*w^2). - Michael Somos, May 29 2005
Given g.f. A(x), then B(q) = q * A(q^8) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2*u6 - u1*u3 * (u2^2 + u6^2). - Michael Somos, May 29 2005
Given g.f. A(x), then B(q) = sqrt(2) * q * A(q^8) satisfies 0 = f(B(q), B(q^7)) where f(u, v) = (1 - u^8) * (1 - v^8) - (1 - u*v)^8. - Michael Somos, Jan 01 2006
Euler transform of period 4 sequence [-1, 2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(-1/2) * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A108494. - Michael Somos, Feb 29 2012
G.f.: Product_{k>0} (1 + x^(2*k)) / (1 + x^(2*k - 1)) = (Sum_{k>0} x^(k^2 - k)) / (Sum_{k>0} x^((k^2 - k)/2)).
G.f.: 1 / (1 + x / (1 + x + x^2 / (1 + x^2 + x^3 / (1 + x^3 + ...)))).
A001935(n) = (-1)^n a(n).
G.f.: (1+1/Q(0))/2, where Q(k)= 1 + x^(k+1) + x^(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013
a(n) ~ (-1)^n * exp(Pi*sqrt(n/2))/(2^(11/4)*n^(3/4)). - Vaclav Kotesovec, Jul 04 2016
G.f.: (-x^2; x^2){-1/2} = ((-1; x^2){1/2})/2, where (a; q)n is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Nov 20 2016
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A109506(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 14 2017
a(n) ~ (-1)^n * exp(Pi*sqrt(n/2)) / (2^(11/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
G.f.: exp(Sum_{k>=1} (-1)^k*x^k/(k*(1 + x^k))). - Ilya Gutkovskiy, May 28 2018

A292929 G.f.: A(x,q) = sqrt( Q(x,q) / Q(x,-q) ), where Q(x,q) = Sum_{n=-oo..+oo} (x - q^n)^n.

Original entry on oeis.org

1, 2, -2, 2, -4, 2, 2, -4, 6, -4, 2, -4, 8, -12, 6, 2, -4, 8, -14, 16, -8, 2, -4, 8, -12, 18, -24, 12, 2, -4, 8, -12, 20, -36, 38, -16, 2, -4, 8, -12, 24, -44, 56, -52, 22, 2, -4, 8, -12, 24, -40, 52, -74, 74, -30, 2, -4, 8, -12, 24, -32, 38, -76, 116, -104, 40, 2, -4, 8, -12, 24, -32, 48, -96, 136, -164, 142, -52, 2, -4, 8, -12, 24, -32, 64, -124, 138, -164, 224, -192, 68, 2, -4, 8, -12, 24, -32, 64, -100, 86, -134, 252, -324, 258, -88, 2, -4, 8, -12, 24, -32, 64, -68, 32, -148, 316, -396, 442, -340, 112, 2, -4, 8, -12, 24, -32, 64, -68, 88, -276, 398, -384, 482, -592, 446, -144, 2, -4, 8, -12, 24, -32, 64, -68, 152, -376, 328, -192, 384, -684, 808, -584, 182, 2, -4, 8, -12, 24, -32, 64, -68, 152, -248, 24, -22, 462, -790, 990, -1074, 752, -228, 2, -4, 8, -12, 24, -32, 64, -68, 152, -120, -152, -288, 1048, -1064, 982, -1272, 1410, -964, 286, 2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 136, -988, 1402, -708, 548, -1168, 1748, -1860, 1232, -356

Offset: 0

Paul D. Hanna, Oct 22 2017

Compare to the g.f. of A108494: sqrt( theta_4(q) / theta_4(-q) ).
Note the related identities:
(1) Sum_{n=-oo..+oo} (x - q^n)^(n-1) = 0.
(2) Sum_{n=-oo..+oo} (x - q^n)^(n+1) = x * Sum_{n=-oo..+oo} (x - q^n)^n.
(3) Sum_{n=-oo..+oo} (x - q^n)^n = 1/(1-x) + Sum_{n>=1} (-1)^n * q^(n^2) * (2 - x*q^n)/(1 - x*q^n)^(n+1).

			G.f.: A(x,q) = Sum_{n>=0} x^n * Sum_{k>=0} T(n,k) * q^(n+k), where
A(x,q) = sqrt( Q(x,q) / Q(x,-q) ) and Q(x,q) is the g.f. of A293600:
Q(x,q) = (1 - 2*q + 2*q^4 - 2*q^9 + 2*q^16 - 2*q^25 + 2*q^36 +...)
+ x*(1 - 3*q^2 + 5*q^6 - 7*q^12 + 9*q^20 - 11*q^30 + 13*q^42 +...)
+ x^2*(1 - 4*q^3 + 9*q^8 - 16*q^15 + 25*q^24 - 36*q^35 + 49*q^48 +...)
+ x^3*(1 - 5*q^4 + 14*q^10 - 30*q^18 + 55*q^28 - 91*q^40 + 140*q^54 +...)
+ x^4*(1 - 6*q^5 + 20*q^12 - 50*q^21 + 105*q^32 - 196*q^45 + 336*q^60 +...)
+ x^5*(1 - 7*q^6 + 27*q^14 - 77*q^24 + 182*q^36 - 378*q^50 + 714*q^66 +...)
+ x^6*(1 - 8*q^7 + 35*q^16 - 112*q^27 + 294*q^40 - 672*q^55 + 1386*q^72 +...)
+ x^7*(1 - 9*q^8 + 44*q^18 - 156*q^30 + 450*q^44 - 1122*q^60 + 792*q^78 +...)
+ ...
Explicitly, the g.f. of this table begins:
A(x,q) = (1 - 2*q + 2*q^2 - 4*q^3 + 6*q^4 - 8*q^5 + 12*q^6 - 16*q^7 + 22*q^8 - 30*q^9 + 40*q^10 - 52*q^11 + 68*q^12 - 88*q^13 +...)
+ x*(2*q - 4*q^2 + 6*q^3 - 12*q^4 + 16*q^5 - 24*q^6 + 38*q^7 - 52*q^8 + 74*q^9 - 104*q^10 + 142*q^11 - 192*q^12 + 258*q^13 - 340*q^14 +...)
+ x^2*(2*q^2 - 4*q^3 + 8*q^4 - 14*q^5 + 18*q^6 - 36*q^7 + 56*q^8 - 74*q^9 + 116*q^10 - 164*q^11 + 224*q^12 - 324*q^13 + 442*q^14 - 592*q^15 +...)
+ x^3*(2*q^3 - 4*q^4 + 8*q^5 - 12*q^6 + 20*q^7 - 44*q^8 + 52*q^9 - 76*q^10 + 136*q^11 - 164*q^12 + 252*q^13 - 396*q^14 + 482*q^15 - 684*q^16 +...)
+ x^4*(2*q^4 - 4*q^5 + 8*q^6 - 12*q^7 + 24*q^8 - 40*q^9 + 38*q^10 - 96*q^11 + 138*q^12 - 134*q^13 + 316*q^14 - 384*q^15 + 384*q^16 - 790*q^17 +...)
+ x^5*(2*q^5 - 4*q^6 + 8*q^7 - 12*q^8 + 24*q^9 - 32*q^10 + 48*q^11 - 124*q^12 + 86*q^13 - 148*q^14 + 398*q^15 - 192*q^16 + 462*q^17 - 1064*q^18 +...)
+ x^6*(2*q^6 - 4*q^7 + 8*q^8 - 12*q^9 + 24*q^10 - 32*q^11 + 64*q^12 - 100*q^13 + 32*q^14 - 276*q^15 + 328*q^16 - 22*q^17 + 1048*q^18 - 708*q^19 +...)
+ x^7*(2*q^7 - 4*q^8 + 8*q^9 - 12*q^10 + 24*q^11 - 32*q^12 + 64*q^13 - 68*q^14 + 88*q^15 - 376*q^16 + 24*q^17 - 288*q^18 + 1402*q^19 + 936*q^20 +...)
+ x^8*(2*q^8 - 4*q^9 + 8*q^10 - 12*q^11 + 24*q^12 - 32*q^13 + 64*q^14 - 68*q^15 + 152*q^16 - 248*q^17 - 152*q^18 - 988*q^19 + 554*q^20 + 1554*q^21 +...)
+ x^9*(2*q^9 - 4*q^10 + 8*q^11 - 12*q^12 + 24*q^13 - 32*q^14 + 64*q^15 - 68*q^16 + 152*q^17 - 120*q^18 + 136*q^19 - 1276*q^20 - 1016*q^21 - 912*q^22+...)
+ x^10*(2*q^10 - 4*q^11 + 8*q^12 - 12*q^13 + 24*q^14 - 32*q^15 + 64*q^16 - 68*q^17 + 152*q^18 - 120*q^19 + 392*q^20 - 636*q^21 - 1432*q^22 - 4352*q^23 +...)
+ x^11*(2*q^11 - 4*q^12 + 8*q^13 - 12*q^14 + 24*q^15 - 32*q^16 + 64*q^17 - 68*q^18 + 152*q^19 - 120*q^20 + 392*q^21 - 124*q^22 - 24*q^23 - 4800*q^24+...)
+ x^12*(2*q^12 - 4*q^13 + 8*q^14 - 12*q^15 + 24*q^16 - 32*q^17 + 64*q^18 - 68*q^19 + 152*q^20 - 120*q^21 + 392*q^22 - 124*q^23 + 1000*q^24 - 1728*q^25 +...)
+ ...
G.F. OF ROWS.
The coefficient of x^0 in A(x,q) is
(R0) Product_{n>=1} (1 - q^(2*n-1)) / (1 + q^(2*n-1)).
The coefficient of x in A(x,q) is
(R1) 2*q * Product_{n>=1} (1 + q^(2*n))/((1 + q^n)*(1 + q^(2*n-1))*(1 + q^(4*n))).
RECTANGULAR ARRAY.
This table of coefficients T(n,k) of x^n*y^(n+k) in A(x,q) begins:
[1, -2, 2, -4, 6, -8, 12, -16, 22, -30, 40, -52, 68, -88, 112, -144, ...];
[2, -4, 6, -12, 16, -24, 38, -52, 74, -104, 142, -192, 258, -340, 446, ...];
[2, -4, 8, -14, 18, -36, 56, -74, 116, -164, 224, -324, 442, -592, 808, ...];
[2, -4, 8, -12, 20, -44, 52, -76, 136, -164, 252, -396, 482, -684, 990, ...];
[2, -4, 8, -12, 24, -40, 38, -96, 138, -134, 316, -384, 384, -790, 982, ...];
[2, -4, 8, -12, 24, -32, 48, -124, 86, -148, 398, -192, 462, -1064, 548, ...];
[2, -4, 8, -12, 24, -32, 64, -100, 32, -276, 328, -22, 1048, -708, -220, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 88, -376, 24, -288, 1402, 936, 1146, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -248, -152, -988, 554, 1554, 5628, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 136, -1276, -1016, -912, 6428, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -636, -1432, -4352, -320, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, -24, -4800, -7696, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, -1728, -7696, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, 320, -1040, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, 320, 3056, ...];
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, 320, 3056, 2836, ...]; ...
The limit of the rows approach A293601, which begins:
[2, -4, 8, -12, 24, -32, 64, -68, 152, -120, 392, -124, 1000, 320, 3056, 2836, 10280, 15112, 38668, 68348, 154152, 297948, 633352, 1269884, 2649892, ...].
RATIOS OF ROW G.F.
The ratios of the row generating functions are as follows.
2 + 2*q^2 + 2*q^6 + 2*q^8 + 2*q^10 + 2*q^12 + 2*q^14 +...
1 + q^2 + q^3 - 3*q^5 + q^6 + 4*q^7 + q^8 - 3*q^9 + q^10 + 3*q^11 +...
1 + q^3 + 3*q^4 - 2*q^5 - 11*q^6 - 3*q^7 + 25*q^8 + 29*q^9 - 33*q^10 +...
1 + 2*q^4 + 6*q^5 - 3*q^6 - 28*q^7 - 27*q^8 + 39*q^9 + 160*q^10 +...
1 + 4*q^5 + 13*q^6 - 4*q^7 - 62*q^8 - 85*q^9 + 19*q^10 + 334*q^11 +...
1 + 8*q^6 + 28*q^7 - 3*q^8 - 134*q^9 - 219*q^10 - 43*q^11 + 571*q^12 +...
1 + 16*q^7 + 60*q^8 + 6*q^9 - 284*q^10 - 557*q^11 - 229*q^12 + 1264*q^13 +...
1 + 32*q^8 + 128*q^9 + 40*q^10 - 590*q^11 - 1380*q^12 - 875*q^13 +...
1 + 64*q^9 + 272*q^10 + 144*q^11 - 1201*q^12 - 3347*q^13 - 2866*q^14 +...
1 + 128*q^10 + 576*q^11 + 432*q^12 - 2392*q^13 - 7966*q^14 - 8598*q^15 +...
1 + 256*q^11 + 1216*q^12 + 1184*q^13 - 4648*q^14 - 18642*q^15 +...
...

Paul D. Hanna, Table of n, a(n) for n = 0..5150 as read by antidiagonals 0..100 of a rectangular array.

Cf. A293600, A293601, A108494 (row 0), A293132 (row 1), A294065 (row 2), A294066 (row 3), A294067 (row 4).

Antidiagonal sums equal zero after the initial '1'.
G.f. of Row 0: Product_{n>=1} (1 - q^(2*n-1)) / (1 + q^(2*n-1)); see A108494.
G.f. of Row 1: 2*q * Product_{n>=1} (1 + q^(2*n))/((1 + q^n)*(1 + q^(2*n-1))*(1 + q^(4*n))).

A193863 Expansion of Product_{n>=0} (1 + q(-q^2)^n) / (1 - q(-q^2)^n).

Original entry on oeis.org

1, 2, 2, 0, -2, 0, 4, 4, -2, -6, 0, 8, 4, -8, -8, 8, 14, -4, -18, 0, 24, 8, -28, -20, 28, 34, -24, -48, 16, 64, 0, -76, -18, 88, 44, -96, -78, 96, 116, -88, -160, 68, 208, -32, -252, -16, 296, 84, -332, -170, 354, 272, -360, -392, 344, 528, -296, -672, 216, 824, -96, -976, -72, 1116, 286, -1240, -552, 1336, 876, -1384

Offset: 0

Joerg Arndt, Aug 07 2011

sign

Expansion of E(-q^2, +q) for E(q,x) = Product_{n>=0} ( 1 + x*q^n ) / ( 1 - x*q^n ).

Replacing q by -q in the g.f. gives the inverse of the g.f., whose expansion is obtained by negating every second term.

			1 + 2*x + 2*x^2 - 2*x^4 + 4*x^6 + 4*x^7 - 2*x^8 - 6*x^9 + 8*x^11 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. Arndt, Regularized q-exponential and q-trigonometric series of the first kind

Cf. A015128 E(+q,+q), A002448 E(+q,-q), A000122 E(-q,+q), A004402 E(-q,-q), A080054 E(+q^2,+q), A108494 E(+q^2,-q), A300574, A300575.

N=66; q='q+O('q^N); /* that many terms */
gf = prod(n=0, N, (1+q*(-q^2)^n)/(1-q*(-q^2)^n) );
Vec(gf) /* show terms */
/* Alternative computation of the g.f. using a product form */
V=[0,-2, 1, 2, 0, -2, -1, 2]; /* note vectors are one-based */
gf=prod(n=0, N, (1-q^n)^(V[n%8+1]) );

{a(n) = local(A); if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k + x * O(x^n) )^[ 0, -2, 1, 2, 0, -2, -1, 2][k%8 + 1]), n))} /* Michael Somos, Feb 26 2012 */

Euler transform of period 8 sequence [ 2, -1, -2, 0, 2, 1, -2, 0, ...]. - Michael Somos, Feb 26 2012
G.f.: prod(n>=0, (1+q*(-q^2)^n)/(1-q*(-q^2)^n) ).
G.f.: sum(n>=0, prod(k=0..n-1, 1+(-q^2)^k )/prod(k=1..n, 1-(-q^2)^k ) * q^n ).
G.f.: sum(n>=0, prod(k=0..n-1, 1+(-q^2)^k)/( prod(k=1..n, 1-(-q^2)^k) * prod(k=0..n-1, 1-q*(-q^2)^k ) ) * q^n * (-q^2)^(n*(n-1)/2) ).
Convolution of A300574 and A300575. - Seiichi Manyama, Nov 22 2019

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A080054 G.f.: Product_{n >= 0} (1+x^(2n+1))/(1-x^(2n+1)).

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A083365 Expansion of psi(x) / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions.

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A292929 G.f.: A(x,q) = sqrt( Q(x,q) / Q(x,-q) ), where Q(x,q) = Sum_{n=-oo..+oo} (x - q^n)^n.

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A193863 Expansion of Product_{n>=0} (1 + q*(-q^2)^n) / (1 - q*(-q^2)^n).

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A193863 Expansion of Product_{n>=0} (1 + q(-q^2)^n) / (1 - q(-q^2)^n).