A083650 -id:A083650 - cp's OEIS frontend

A030204 Expansion of q^(-1/8) * eta(q) * eta(q^2) in powers of q.

1, -1, -2, 1, 0, 2, 1, 0, 0, -2, 1, -2, -2, 0, 2, -1, 0, 2, 0, 2, 0, 1, 0, 0, -2, 0, 0, 0, -1, -2, -2, 0, 2, 0, 0, -2, 3, 0, 0, 2, 0, 0, 2, 0, 2, -1, -2, 0, 0, 0, -2, 2, 0, -2, -2, -1, -2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 2, 2, 0, 2, -2, 0, -2, 1, 0

Offset: 0

N. J. A. Sloane

sign

Number 66 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
A030204, A083650 and A138514 are the same except for signs. - N. J. A. Sloane, May 07 2010

			G.f. = 1 - x - 2*x^2 + x^3 + 2*x^5 + x^6 - 2*x^9 + x^10 - 2*x^11 - 2*x^12 + ...
G.f. = q - q^9 - 2*q^17 + q^25 + 2*q^41 + q^49 - 2*q^73 + q^81 - 2*q^89 - 2*q^97 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334. [a(n) = coefficient of q^(9m+1) in the q-expansion of the unique normalized newform g of weight 1, level 128, and character chi_2. - _N. J. A. Sloane_, Oct 18 2014]
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000203, A002513, A107063, A107064, A138514.

Basis( CuspForms( Gamma1(128), 1), 641)[1]; /* Michael Somos, Jan 31 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, I x] / (4 Sqrt[ x] I^(1/4)), {x, 0, 4 n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, Pi/4, x] / (2^(3/2) x^(1/2)), {x, 0, 4 n}]; (* Michael Somos, Jan 31 2015 *)

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

{a(n) = my(A, p, e); if( n<0, 0, n = 8*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%8==1, (e + 1) * if( qfbclassno(-4*p)%8, (-1)^e, 1), e%2==0, (-1)^(e/2*(p%8<5)))))}; /* Michael Somos, Jul 26 2006 */

{a(n) = if( n<0, 0, n = 8*n + 1; (qfrep([1, 0;0, 32], n) - qfrep([4, 2; 2, 9], n))[n])}; /* Michael Somos, Sep 02 2006 */

G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k)).
G.f.: (Sum_{k>0} x^((k^2 - k)/2)) * (Sum_{k in Z} (-1)^k * x^k^2). - Michael Somos, Sep 02 2006
Expansion of psi(x) * phi(-x) = f(-x^2) * f(-x) = f(-x)^2 / chi(-x) = f(-x)^3 / phi(-x) = f(-x^2)^2 * chi(-x) = f(-x^2)^3 / psi(x) = psi(-x) * phi(-x^2) = psi(x)^2 * chi(-x)^3 = phi(-x)^2 / chi(-x)^3 = (f(-x)^3 * psi(x))^(1/2) = (f(-x^2)^3 * phi(-x))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Mar 22 2008
Expansion of psi(x) * psi(-x) in powers of x^2 where psi() is a Ramanujan theta function. - Michael Somos, Oct 11 2013
Euler transform of period 2 sequence [ -1, -2, ...].
a(3*n) = A107063(n). a(3*n + 2) = -2 * A107064(n). - Michael Somos, Oct 11 2013
a(9*n + 1) = -a(n), a(9*n + 4) = a(9*n + 7) = 0. - Michael Somos, Mar 17 2004
a(n) = b(8*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 0 if p === 3,5,7 (mod 8) and e odd, b(p^e) = (-1)^(e/2) if p == 3 (mod 8) and e even, b(p^e) = 1 if p == 5,7 (mod 8) and e even, b(p^e) = e + 1 if p == 1 (mod 8) and p = x^2 + 32*y^2, b(p^e) = (-1)^e * (e + 1) if p == 1 (mod 8) and p is not of the form x^2 + 32*y^2.
a(n) = (-1)^n * A138514(n). Convolution inverse is A002513.
G.f.: exp(Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

A170925 G.f.: eta(q)eta(q^2)eta(q^4)eta(q^8)eta(q^16)eta(q^32)..., where eta(q) = Product((1-q^m), m=1..oo).

Original entry on oeis.org

1, -1, -2, 1, -1, 3, 3, -1, -2, -2, 4, -4, -1, -3, -3, 2, 1, 9, -1, 6, 7, -8, -10, 1, -1, 0, -2, 0, 2, -1, 4, -4, -1, -5, 14, -15, -7, 9, 11, 7, 0, 3, -14, 17, -7, 18, 4, -6, -7, -25, -12, -5, -13, -3, 9, -14, 25, 10, -2, 8, 17, 1, 2, 13, 4, 0, -4, 7, 13, -27, -42, 11, 5, 5, 10, -24, 3, -21, -4, 0, -32, 27, 29, -1, -4, 43, 26, -7, -41, -9, 27, -11

Offset: 0

N. J. A. Sloane and Gary W. Adamson, Feb 18 2010

sign

eta(q) = A(q)/A(q^2), where A(q) is the g.f. for this sequence (cf. A010815).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A010815, A083650, A030189, A160832, A143374, A194087.

nmax = 100; CoefficientList[Series[Product[QPochhammer[x^(2^k)], {k, 0, Log[nmax]/Log[2]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2019 *)

def s(k, n)
  s = 0
  (1..n).each{|i| s += i if n % i == 0 && i % k == 0}
  s
end
def A(ary, n)
  a_ary = [1]
  a = [0] + (1..n).map{|i| ary.inject(0){|s, j| s + j[1] * s(j[0], i)}}
  (1..n).each{|i| a_ary << (1..i).inject(0){|s, j| s - a[j] * a_ary[-j]} / i}
  a_ary
end
def A170925(n)
  A((0..Math.log(n, 2)).map{|i| [2 ** i, 1]}, n)
end
p A170925(100) # Seiichi Manyama, Sep 23 2019

A143374 G.f.: eta(q)eta(q^3)eta(q^9)eta(q^27)eta(q^81)eta(q^243)..., where eta(q) = Product((1-q^m), m=1..oo).

Original entry on oeis.org

1, -1, -1, -1, 1, 2, -1, 2, 0, -1, 0, 0, 0, -2, -2, 2, -3, -1, 1, 2, 3, 4, 1, -3, 0, -2, 3, -4, 2, 0, -1, -1, -2, -1, 0, -2, -2, 2, 2, -1, 0, 5, -1, 5, 0, 2, -3, -3, -3, 1, 3, 2, 2, -2, 4, -6, -4, 2, -2, -1, 2, -6, 0, 8, -4, -3, 2, 5, 1, -6, 3, 6, -1, 1, -4, -10, 1, 2, -1, 2, -5, -2, 6, 13, 4, 1, -1, 2, 1, 4, -4, -1

Offset: 0

N. J. A. Sloane and Gary W. Adamson, Feb 18 2010, Aug 14 2011

sign

eta(q) = A(q)/A(q^3), where A(q) is the g.f. for this sequence (cf. A010815).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A010815, A083650, A030189, A160832, A170925, A173241, A194087.

A138514 Expansion of q^(-1/8) * eta(q^2)^4 / (eta(q) * eta(q^4)) in powers of q.

Original entry on oeis.org

1, 1, -2, -1, 0, -2, 1, 0, 0, 2, 1, 2, -2, 0, 2, 1, 0, -2, 0, -2, 0, -1, 0, 0, -2, 0, 0, 0, -1, 2, -2, 0, 2, 0, 0, 2, 3, 0, 0, -2, 0, 0, 2, 0, 2, 1, -2, 0, 0, 0, -2, -2, 0, 2, -2, 1, -2, -2, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, -2, 2, 0, 2, 2, 0, 2, 1, 0, -2, 0, 2, 0, -2, 0, 0, 4, 0, 0, 0, 1, 0, 0, 0, -2, -2, 0, 0, 0, 2, -2, 0, 0, -2

Offset: 0

Michael Somos, Mar 22 2008

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 70 of the 74 eta-quotients listed in Table I of Martin (1996).
A030204, A083650 and A138514 are the same except for signs. - N. J. A. Sloane, May 07 2010

			G.f. = 1 + x - 2*x^2 - x^3 - 2*x^5 + x^6 + 2*x^9 + x^10 + 2*x^11 - 2*x^12 + ...
G.f. = q + q^9 - 2*q^17 - q^25 - 2*q^41 + q^49 + 2*q^73 + q^81 + 2*q^89 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A030204, A083650.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ -x], {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^4 / (QPochhammer[ x] QPochhammer[ x^4]), {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)

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

{a(n) = my(A, p, e); if( n<0, 0, n = 8*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%8==1, (e+1) * if( qfbclassno( -8 * p) / 4 % 2, (-1)^e, 1), if( e%2==0, (-1)^(e/2 * (p%8==5)))))) };

{a(n) = if( n<0, 0, n = 8*n + 1; (qfrep([ 1, 0; 0, 64], n) - qfrep([ 4, 2; 2, 17], n))[n])};

Expansion of f(x) * f(-x^2) = psi(-x) * phi(x) = chi(x) * f(-x^2)^2 = psi(x) * phi(-x^2) = f(x)^2 / chi(x) = f(x)^3 / phi(x) = f(-x^2)^3 / psi(-x) = phi(x)^2 / chi(x)^3 = chi(x)^3 * psi(-x)^2 = (f(x)^3 * psi(-x))^(1/2) = (f(-x^2)^3 * phi(x))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of psi(i * x) * psi(-i * x) in powers x^2 where i^2 = -1 and psi() is a Ramanujan theta function. - Michael Somos, Feb 16 2014
Euler transform of period 4 sequence [ 1, -3, 1, -2, ...].
a(n) = b(8*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 0 if p == 3, 5, 7 (mod 8) and e odd, b(p^e) = 1 if p == 3 (mod 4) and e even, b(p^e) = (-1)^(e/2) if p == 5 (mod 8) and e even, b(p^e) = e+1 if p == 1 (mod 8) and p = x^2 + 64*y^2, b(p^e) = (-1)^e * (e+1) if p == 1 (mod 8) and p is not of the form x^2 + 64*y^2.
a(9*n + 1) = a(n), a(9*n + 4) = a(9*n + 7) = 0. a(n) = (-1)^n * A030204(n) = (-1)^floor((n+1)/2) * A083650(n).
G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k - 1)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 16 (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 10 2015

A160832 Expansion of eta(q)eta(q^2)eta(q^4), where eta(q) = Product((1-q^m), m=1..oo).

Original entry on oeis.org

1, -1, -2, 1, -1, 3, 3, -1, -1, -3, 2, -3, -2, 0, 0, 1, 2, 4, -3, 5, 3, -2, -4, 0, -2, -1, 1, -2, 2, -6, -3, -1, 3, 4, 5, -3, 2, 2, 3, 4, -7, 1, 4, -1, -3, 1, -4, 0, -4, 1, -2, 1, -2, -3, 1, -5, 0, 4, 1, 3, 5, 1, 4, -1, 7, -5, -2, 0, 0, -1, -2, 6, 8, -5, -5, -4, -3, 0, -1, 0, -6, -1, -3, 3, -3, 6, -2, -6, 6, 1, -4, 6, 0, 5, 6, 7, -5, -4, 4, -5, 2, 4, 6, -4, -3

Offset: 0

N. J. A. Sloane and Gary W. Adamson, Feb 18 2010

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A010815, A083650, A030189, A170925.

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-7/24)* eta[q]*eta[q^2]*eta[q^4], {q, 0, 100}], q] (* G. C. Greubel, Apr 30 2018 *)

q='q+O('q^50); Vec(eta(q)*eta(q^2)*eta(q^4)) \\ G. C. Greubel, Apr 30 2018

cp's OEIS Frontend

A030204 Expansion of q^(-1/8) * eta(q) * eta(q^2) in powers of q.

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A170925 G.f.: eta(q)*eta(q^2)*eta(q^4)*eta(q^8)*eta(q^16)*eta(q^32)*..., where eta(q) = Product((1-q^m), m=1..oo).

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A143374 G.f.: eta(q)*eta(q^3)*eta(q^9)*eta(q^27)*eta(q^81)*eta(q^243)*..., where eta(q) = Product((1-q^m), m=1..oo).

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A138514 Expansion of q^(-1/8) * eta(q^2)^4 / (eta(q) * eta(q^4)) in powers of q.

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A160832 Expansion of eta(q)*eta(q^2)*eta(q^4), where eta(q) = Product((1-q^m), m=1..oo).

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A194087 G.f.: eta(q)*eta(q^4)*eta(q^16)*eta(q^64)*eta(q^256)*eta(q^1024)*..., where eta(q) = Product_{m=1..oo} (1 - q^m).

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A170925 G.f.: eta(q)eta(q^2)eta(q^4)eta(q^8)eta(q^16)eta(q^32)..., where eta(q) = Product((1-q^m), m=1..oo).

A143374 G.f.: eta(q)eta(q^3)eta(q^9)eta(q^27)eta(q^81)eta(q^243)..., where eta(q) = Product((1-q^m), m=1..oo).

A160832 Expansion of eta(q)eta(q^2)eta(q^4), where eta(q) = Product((1-q^m), m=1..oo).

A194087 G.f.: eta(q)eta(q^4)eta(q^16)eta(q^64)eta(q^256)eta(q^1024)..., where eta(q) = Product_{m=1..oo} (1 - q^m).