A127392 -id:A127392 - cp's OEIS frontend

A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^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, 1545306

Offset: 0

N. J. A. Sloane

The Cayley reference is actually to A079006. - Michael Somos, Feb 24 2011
In the math overflow link is a conjecture that a(n) == a(9*n + 2) (mod 4).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of 4-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019
The g.f. in the form A(x) = Sum_{k >= 0} x^(k*(k+1)) / (1 + 2*Sum_{k >= 1} (-1)^k * x^(k^2)) == Sum_{k >= 0} x^(k*(k+1)) (mod 2). It follows that a(n) is odd iff n = k*(k + 1) for some nonnegative integer k. - Peter Bala, Jan 04 2025

			G.f. = 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. = 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).
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
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.
joro on Math Overflow, Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable mod 4?.
T. Kathiravan and S. N. Fathima, On L-regular bipartitions modulo L, The Ramanujan Journal 44.3 (2017): 549-558.
H. P. Robinson, Letter to N. J. A. Sloane, Oct 07, 1976
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A001935, A022568, A079006, A082304, A098613, A127391, A127392, A210656.

Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548, A333374.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> [2,2,2,0] [modp(n-1,4)+1]): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008
f:=(k,M) -> mul(1-q^(k*j),j=1..M); LRBP := (L,M) -> (f(L,M)/f(1,M))^2; S := L -> seriestolist(series(LRBP(L,80),q,60)); S(4); # N. J. A. Sloane, Oct 20 2019

m = 38; CoefficientList[ Series[ Product[ (1 - x^(4*k))/(1 - x^k), {k, 1, m}]^2 , {x, 0, m}], x] (* Jean-François Alcover, Sep 02 2011, after g.f. *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0, x]) / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - x^k, {k, 4, n, 4}] / Product[ 1 - x^k, {k, n}])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^4] / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x] QPochhammer[ -x^2, x^2])^2, {x, 0, n}]; (* Michael Somos, May 16 2015 *)

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

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, 1 / if(k%4, 1 - x^k, 1), 1 + x * O(x^n))^2, n))};

G.f.: Product ( 1 - x^k )^(-c(k)); c(k) = 2, 2, 2, 0, 2, 2, 2, 0, ....
Convolution square of A001935. A079006(n) = (-1)^n a(n).
Expansion of q^(-1/4) * (1/2) * (k / k')^(1/2) in powers of q.
Euler transform of period 4 sequence [ 2, 2, 2, 0, ...].
Given g.f. A(x), then B(q) = (q * A(q^4))^4 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (1 + 16*u) * (1 + 16*v) * v - u^2. - Michael Somos, Jul 09 2005
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^2 + v^2)^2 - u*v * (1 + 4*u*v)^2. - 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.
Equals A000009 convolved with A098613. - Gary W. Adamson, Mar 24 2011
a(9*n + 2) = a(n) + 4 * A210656(3*n). - Michael Somos, Apr 02 2012
Convolution inverse is A082304. - Michael Somos, May 16 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = (1/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A082304. - Michael Somos, May 16 2015
Expansion of f(-x^4)^2 / f(-x)^2 = psi(x^2) / phi(-x) = psi(-x)^2 / phi(-x)^2 = psi(x)^2 / phi(-x^2)^2 = psi(x^2)^2 / psi(-x)^2 = chi(x)^2 / chi(-x^2)^4 = 1 / (chi(x)^2 * chi(-x)^4) = 1 / (chi(-x)^2 * chi(-x^2)^2) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, May 16 2015
a(n) ~ exp(Pi*sqrt(n)) / (8*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Aug 18 2015
G.f.: A(x) = Sum_{n >= 0} x^(n*(n+1)) / Sum_{n = -oo..oo} (-1)^n*x^(n^2). - Peter Bala, Feb 19 2021

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

Original entry on oeis.org

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

A007096 Expansion of theta_3 / theta_4.

Original entry on oeis.org

1, 4, 8, 16, 32, 56, 96, 160, 256, 404, 624, 944, 1408, 2072, 3008, 4320, 6144, 8648, 12072, 16720, 22976, 31360, 42528, 57312, 76800, 102364, 135728, 179104, 235264, 307672, 400704, 519808, 671744, 864960, 1109904, 1419456, 1809568, 2299832

Offset: 0

N. J. A. Sloane, Simon Plouffe

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of partitions of 2n into parts with 2 types c, c* of each part. The even parts appears with multiplicity 1 for each type. The odd parts appears with multiplicity 2 (cc or c*c* but not cc*, that is, no mixing is allowed). E.g., a(4)=8 because of 44*, 22*, 211, 21*1*, 2*1*1*, 2*11, 111*1*. - Noureddine Chair, Jan 27 2005
a(n) is the number of pairs of overpartitions into odd parts where the sum of all parts is equal to n. - Jeremy Lovejoy, Aug 29 2020

			G.f. = 1 + 4*q + 8*q^2 + 16*q^3 + 32*q^4 + 56*q^5 + 96*q^6 + 160*q^7 + 256*q^8 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 8, 13.
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.
Bernard L.S. Lin, Arithmetic properties of overpartition pairs into odd parts, Electronic J. Combin. 19, 2012, Paper 17.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Self-convolution of A080054. - Vladeta Jovovic, Mar 22 2005
Cf. A014969, A001936, A001938, A079006, A127391, A127392.
Cf. A022577, A080054, A097243, A123655.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] / EllipticTheta[ 4, 0, q], {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ (1 - m)^(-1/4), {q, 0, n}]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[( QPochhammer[ -q, q^2] / QPochhammer[ q, q^2])^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ (Product[ 1 - (-q)^k, {k, n}] / Product[ 1 - q^k, {k, n}])^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^2, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)

{a(n) = my(A, B); if( n<0, 0, A = 1 + 4*x; for( k=2, n, B = A + x^2 * O(x^k); A += Pol(2 * subst(B, x, x^2)^2 - B - 1/B) / x / 8); polcoeff(A, n))}; /* Michael Somos, Jul 07 2005*/

{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)))^2, n))}; /* Michael Somos, Jan 01 2006 */

Euler transform of period 4 sequence [4, -2, 4, 0, ...]. - Vladeta Jovovic, Mar 22 2005
Expansion of eta(q^2)^6 /(eta(q)^4 * eta(q^4)^2) in powers of q.
Expansion of phi(q) / phi(-q) = chi(q)^2 / chi(-q)^2 = psi(q)^2 / psi(-q)^2 = phi(-q^2)^2 / phi(-q)^2 = phi(q)^2 / phi(-q^2)^2 = chi(-q^2)^2 / chi(-q)^4 = chi(q)^4 / chi(-q^2)^2 = f(q)^2 / f(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
G.f. A(x) satisfies 0 = f(A(x), A(x^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 A028939.
Expansion of Jacobian elliptic function 1 / sqrt(k') in powers of q. - see Fine.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = 1 + u^2 - 2*u*v^2. - Michael Somos, Jul 07 2005
Unique solution to f(x^2)^2 = (f(x) + 1 / f(x)) / 2 and f(0)=1, f'(0) nonzero.
G.f.: theta_3 / theta_4 = (Sum_{k} x^k^2) / (Sum_{k} (-x)^k^2) = (Product_{k>0} (1 - x^(4*k - 2)) / ((1 - x^(4*k - 1)) * (1 - x^(4*k - 3)))^2)^2.
A097243(n) = a(4*n). 8*A022577(n) = a(4*n + 2). a(n) = 4*A123655(n) if n>0. Convolution square of A080054.
Empirical: sum(exp(-Pi)^(n-1)*a(n),n=1..infinity) = 2^(1/4). - Simon Plouffe, Feb 20 2011
Empirical : sum(exp(-Pi*sqrt(2))^(n-1)*(-1)^(n+1)*a(n),n=1..infinity) = (-2+2*2^(1/2))^(1/4). - Simon Plouffe, Feb 20 2011
Empirical : sum(exp(-2*Pi)^(n-1)*a(n),n=1..infinity) = 1/2*(8+6*2^(1/2))^(1/4). - Simon Plouffe, Feb 20 2011
a(n) ~ exp(Pi*sqrt(n)) / (4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Aug 28 2015
G.f.: exp(4*Sum_{k>=1} sigma(2*k - 1)*x^(2*k-1)/(2*k - 1)). - Ilya Gutkovskiy, Apr 19 2019

A127391 Series expansion of the elliptic function sqrt(k) = theta_2/theta_3 in powers of q^(1/4).

Original entry on oeis.org

0, 2, 0, 0, 0, -4, 0, 0, 0, 10, 0, 0, 0, -20, 0, 0, 0, 36, 0, 0, 0, -64, 0, 0, 0, 110, 0, 0, 0, -180, 0, 0, 0, 288, 0, 0, 0, -452, 0, 0, 0, 692, 0, 0, 0, -1044, 0, 0, 0, 1554, 0, 0, 0, -2276, 0, 0, 0, 3296, 0, 0, 0, -4724, 0, 0, 0, 6696, 0, 0, 0, -9408, 0, 0, 0, 13108, 0, 0, 0, -18112, 0, 0, 0, 24850, 0

Offset: 0

N. J. A. Sloane, Mar 31 2007

sign

It appears that a(n) = 2 * A208933(n) - A212318(n) for n>0. - Thomas Baruchel, May 14 2018

Empirical: Sum_{n>=1} a(n)/exp(Pi*(n-1)) = 3 + 2*sqrt(2) - 2*sqrt(4 + 3*sqrt(2)). - Simon Plouffe, Mar 01 2021

			2*x - 4*x^5 + 10*x^9 - 20*x^13 + 36*x^17 - 64*x^21 + 110*x^25 -180*x^29 + ...
2*q^(1/4) - 4*q^(5/4) + 10*q^(9/4) - 20*q^(13/4) + 36*q^(17/4) - 64*q^(21/4) + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

See A127392 for another version. Dividing by 2 gives A079006. Cf. A001936, A001938.

A161800 G.f.: A(q) = exp( Sum_{n>=1} A002129(n) * 2A006519(n) q^n/n ).

Original entry on oeis.org

1, 2, 0, 0, -6, -16, 0, 0, -8, 18, 0, 0, 112, 176, 0, 0, -86, -544, 0, 0, -752, -160, 0, 0, 1360, 2834, 0, 0, 1216, -5104, 0, 0, -5384, 3232, 0, 0, 10762, 18032, 0, 0, -8176, -68992, 0, 0, -59888, 48400, 0, 0, 130160, 143074, 0, 0, 47696, -343088, 0, 0

Offset: 0

Paul D. Hanna, Jul 19 2009

sign

A002129 forms the l.g.f. of log[ Sum_{n>=0} q^(n(n+1)/2) ], while
2*A006519 forms the l.g.f. of binary partitions (A000123) and
A006519(n) is the highest power of 2 dividing n.

			G.f.: A(q) = 1 + 2*q - 6*q^4 - 16*q^5 - 8*q^8 + 18*q^9 + 112*q^12 + 176*q^13 +...
log(A(q)) = 2*q - 4*q^2/2 + 8*q^3/3 - 40*q^4/4 + 12*q^5/5 - 16*q^6/6 +...
Sum_{n>=1} A002129(n)*q^n/n = log(1 + q + q^3 + q^6 + q^10 + q^15 +...),
Sum_{n>=1} 2*A006519(n)*x^n/n = log of the g.f. of binary partitions A000123.
QUADRASECTIONS:
Q_0(q) = 1 - 6*q - 8*q^2 + 112*q^3 - 86*q^4 - 752*q^5 + 1360*q^6 +...
Q_1(q) = 2 - 16*q + 18*q^2 + 176*q^3 - 544*q^4 - 160*q^5 + 2834*q^6 +...
The ratio Q_1(q)/Q_0(q) yields:
2 - 4*q + 10*q^2 - 20*q^3 + 36*q^4 - 64*q^5 + 110*q^6 - 180*q^7 +...
which appears to equal the g.f. of A127392.

Cf. A127392, quadrasections: A161801, A161802.

Cf. A002129, A006519, A000123.

{a(n)=local(L=sum(m=1, n,2*2^valuation(m,2)*sumdiv(m, d, -(-1)^d*d)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}

a(n) = 0 when n == 2 or 3 (mod 4).
Define the nonzero series QUADRASECTIONS:
Q_0(q) = Sum_{n>=0} a(4n)*q^n,
Q_1(q) = Sum_{n>=0} a(4n+1)*q^n, then:
Q_1(q)/Q_0(q) = series expansion of the elliptic function sqrt(k)/q^(1/4), where sqrt(k) = theta_2/theta_3, as described by A127392.
[The above statements are conjectures needing proof.]

cp's OEIS Frontend

A001936 Expansion of q^(-1/4) * (eta(q^4) / eta(q))^2 in powers of q.

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

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A007096 Expansion of theta_3 / theta_4.

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A127391 Series expansion of the elliptic function sqrt(k) = theta_2/theta_3 in powers of q^(1/4).

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A161800 G.f.: A(q) = exp( Sum_{n>=1} A002129(n) * 2*A006519(n) * q^n/n ).

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A161800 G.f.: A(q) = exp( Sum_{n>=1} A002129(n) * 2A006519(n) q^n/n ).