A113416 -id:A113416 - cp's OEIS frontend

A113418 Expansion of (eta(q^2)^7eta(q^4)/(eta(q)eta(q^8))^2-1)/2 in powers of q.

1, -1, -2, -1, -4, 2, 8, -1, 7, 4, -10, 2, -12, -8, 8, -1, 18, -7, -18, 4, -16, 10, 24, 2, 21, 12, -20, -8, -28, -8, 32, -1, 20, -18, -32, -7, -36, 18, 24, 4, 42, 16, -42, 10, -28, -24, 48, 2, 57, -21, -36, 12, -52, 20, 40, -8, 36, 28, -58, -8, -60, -32, 56, -1, 48, -20, -66, -18, -48, 32, 72, -7, 74, 36, -42, 18, -80, -24

Offset: 1

Michael Somos, Oct 29 2005

Amiram Eldar, Table of n, a(n) for n = 1..10000

Apart from signs, same as A117000.

A113416(n)=2*a(n) if n>0.

f[p_, e_] := If[1 < Mod[p, 8] < 7, ((-p)^(e+1)-1)/(-p-1), (p^(e+1)-1)/(p-1)]; f[2, e_] := -1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 22 2023 *)

a(n)=if(n<1, 0, -sumdiv(n,d, d*(d%2)*(-1)^(n/d+(d+1)\4)))

{a(n)=local(A,p,e); if(n<1, 0, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==2, -1, p*=kronecker(2,p); (p^(e+1)-1)/(p-1)))))}

a(n) is multiplicative and a(2^e) = -1 if e>0, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 7 (mod 8), a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 5 (mod 8).
G.f.: Sum_{k>0} (2k-1)*(-1)^[k/2]*x^(2k-1)/(1+x^(2k-1)).
From Amiram Eldar, Jan 28 2024: (Start)
a(n) = (-1)^(n+1) * A117000(n).
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(2)) = 0.290786... . (End)

A131999 Expansion of eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2 in powers of q.

Original entry on oeis.org

1, -2, -2, 4, -2, 8, 4, -16, -2, -14, 8, 20, 4, 24, -16, -16, -2, -36, -14, 36, 8, 32, 20, -48, 4, -42, 24, 40, -16, 56, -16, -64, -2, -40, -36, 64, -14, 72, 36, -48, 8, -84, 32, 84, 20, 56, -48, -96, 4, -114, -42, 72, 24, 104, 40, -80, -16, -72, 56, 116, -16

Offset: 0

Michael Somos, Aug 06 2007

sign

Number 19 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).

			G.f. = 1 - 2*q - 2*q^2 + 4*q^3 - 2*q^4 + 8*q^5 + 4*q^6 - 16*q^7 - 2*q^8 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.67).

G. C. Greubel, Table of n, a(n) for n = 0..10000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
Yves 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. A113416, A113417, A117000, A124340, A259491.

A := Basis( ModularForms( Gamma1(8), 2), 61); A[1] - 2*A[2] - 2*A[3] + 4*A[4] - 2*A[5]; /* Michael Somos, Jun 28 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], -2 DivisorSum[ n, # KroneckerSymbol[ 2, #] &]]; (* Michael Somos, Jun 28 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^3 / QPochhammer[ q^8]^2, {q, 0, n}]; (* Michael Somos, Jun 28 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q]^2, {q, 0, n}]; (* Michael Somos, Jun 28 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^4]^2, {q, 0, n}]; (* Michael Somos, Jun 28 2015 *)

{a(n) = if( n<1, n==0, -2 * sumdiv(n, d, d * kronecker( 2, d)))};

{a(n) = my(A, p, e); if( n<0, n==0, A = factor(n); -2 * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 1, abs(p%8-4)==3, (p^(e+1) - 1) / (p - 1), ((-p)^(e+1) - 1) / (-p - 1))))};

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

Expansion of phi(q) * phi(q^2) * phi(-q)^2 in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 8 sequence [-2, -3, -2, -6, -2, -3, -2, -4, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^4 + u^2*v^2 + 2 * u^2*w^2 + 2 * u*v*w * (-u + 2*v - 2*w) - 2 * u*v^3.
a(n) = 2 * b(n) where b() is multiplicative with b(2^e) = 1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 7 (mod 8), b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 5 (mod 8).
a(2*n) = a(n) for all n in Z.
G.f.: 1 - 2* Sum_{k>0} k * x^k / (1 - x^k) * Kronecker(2, k).
G.f.: Product_{k>0} (1 - x^k)^4 * (1 + x^k)^2 * (1 + x^(2*k)) / (1 + x^(4*k))^2.
a(n) = -2 * A117000(n) unless n=0. a(n) = (-1)^n * A113416(n). a(2*n + 1) = - 2 * A113417(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(11/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A124340. - Michael Somos, Jun 28 2015
Convolution square is A259491. - Michael Somos, Jun 28 2015

cp's OEIS Frontend

A113418 Expansion of (eta(q^2)^7eta(q^4)/(eta(q)eta(q^8))^2-1)/2 in powers of q.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A131999 Expansion of eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2 in powers of q.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

cp's OEIS Frontend

A113418 Expansion of (eta(q^2)^7*eta(q^4)/(eta(q)*eta(q^8))^2-1)/2 in powers of q.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A131999 Expansion of eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2 in powers of q.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

A113418 Expansion of (eta(q^2)^7eta(q^4)/(eta(q)eta(q^8))^2-1)/2 in powers of q.