A030189 -id:A030189 - cp's OEIS frontend

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).

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.

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

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).

Original entry on oeis.org

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

Offset: 0

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

sign

eta(q) = A(q)/A(q^4), 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, A143374.

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Ruby

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).

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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).

Views

Author

Keywords

Comments

Links

Crossrefs

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Ruby

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).

Views

Author

Keywords

Comments

Links

Crossrefs

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

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).

Views

Author

Keywords

Comments

Links

Crossrefs

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).