A113417 -id:A113417 - cp's OEIS frontend

A113419 Expansion of phi(x)^2 * phi(-x) * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

1, 2, -4, -8, 7, 10, -12, -8, 18, 18, -16, -24, 21, 20, -28, -32, 20, 32, -36, -24, 42, 42, -28, -48, 57, 36, -52, -40, 36, 58, -60, -56, 48, 66, -48, -72, 74, 42, -80, -80, 61, 82, -72, -56, 90, 96, -64, -72, 98, 70, -100, -104, 64, 106, -108, -72, 114, 96

Offset: 0

Michael Somos, Oct 29 2005

sign

Number 46 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*x - 4*x^2 - 8*x^3 + 7*x^4 + 10*x^5 - 12*x^6 - 8*x^7 + 18*x^8 + ...
G.f. = q + 2*q^3 - 4*q^5 - 8*q^7 + 7*q^9 + 10*q^11 - 12*q^13 - 8*q^15 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
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. A109039, A113417, A209940, A258096.

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^9 QPochhammer[ q^8]^2 / (QPochhammer[ q]^2 QPochhammer[ q^4]^5), {q, 0, n}]; (* Michael Somos, May 19 2015 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, KroneckerSymbol[ 2, #] # &]]; (* Michael Somos, May 19 2015 *)

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

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^5), n))};

Expansion of q^(-1/2) * (eta(q^2)^9 * eta(q^8)^2) / (eta(q)^2 * eta(q^4)^5) in powers of q. - Michael Somos, Mar 14 2012
Euler transform of period 8 sequence [ 2, -7, 2, -2, 2, -7, 2, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (x^(e+1) - y^(e+1)) / (x - y) where x = p * (-1)^floor(p/4) and y = (-1)^floor(p/2).
G.f.: Sum_{k>0} (2*k - 1) * (-1)^[(k - 1)/2] * x^(2*k - 1) / (1 + x^(4*k - 2)).
a(n) = (-1)^n * A113417(n) = (-1)^floor(n/2) * A258096(n) = (-1)^(n + floor(n/2)) * A209940(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109039. - Michael Somos, May 20 2015

A209940 Expansion of psi(x^4) * phi(-x^4)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta function.

Original entry on oeis.org

1, -2, 4, -8, 7, -10, 12, -8, 18, -18, 16, -24, 21, -20, 28, -32, 20, -32, 36, -24, 42, -42, 28, -48, 57, -36, 52, -40, 36, -58, 60, -56, 48, -66, 48, -72, 74, -42, 80, -80, 61, -82, 72, -56, 90, -96, 64, -72, 98, -70, 100, -104, 64, -106, 108, -72, 114, -96

Offset: 0

Michael Somos, Mar 16 2012

sign

Number 47 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*x + 4*x^2 - 8*x^3 + 7*x^4 - 10*x^5 + 12*x^6 - 8*x^7 + 18*x^8 + ...
G.f. = q - 2*q^3 + 4*q^5 - 8*q^7 + 7*q^9 - 10*q^11 + 12*q^13 - 8*q^15 + 18*q^17 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 44.
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. A113417, A113419, A258096.

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^4]^9 / (QPochhammer[ q^2]^5 QPochhammer[ q^8]^2), {q, 0, n}]; (* Michael Somos, May 19 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^9 / (eta(x^2 + A)^5 * eta(x^8 + A)^2), n))};

{a(n) = my(A, p, e, f); if( n<0, 0, A = factor(2*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0^e, p==3, ((-p)^(e+1) - 1) / ((-p) - 1), p *= kronecker( 18, p); (-1)^(e*(p\6)) * (p^(e+1) - 1) / (p - 1))))};

Expansion of q^(-1/2) * eta(q)^2 * eta(q^4)^9 / (eta(q^2)^5 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, 3, -2, -6, -2, 3, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 512^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113419.
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = (1 - (-3)^(e+1)) / 4, b(p^e) = (-1)^(e * [p/6]) * ((p*f)^(e+1) - 1) / (p*f - 1) where f = Kronecker( 18, p).
a(n) = (-1)^n * A258096(n) = (-1)^floor(n/2) * A113419(n) = (-1)^(n + floor(n/2)) * A113417(n).

A258096 Expansion of psi(x^4) * phi(-x^4)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta function.

Original entry on oeis.org

1, 2, 4, 8, 7, 10, 12, 8, 18, 18, 16, 24, 21, 20, 28, 32, 20, 32, 36, 24, 42, 42, 28, 48, 57, 36, 52, 40, 36, 58, 60, 56, 48, 66, 48, 72, 74, 42, 80, 80, 61, 82, 72, 56, 90, 96, 64, 72, 98, 70, 100, 104, 64, 106, 108, 72, 114, 96, 84, 144, 111, 84, 104, 128

Offset: 0

Michael Somos, May 19 2015

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 7*x^4 + 10*x^5 + 12*x^6 + 8*x^7 + ...
G.f. = q + 2*q^3 + 4*q^5 + 8*q^7 + 7*q^9 + 10*q^11 + 12*q^13 + 8*q^15 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A113417, A113419, A209940.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ x^4]^7 / (QPochhammer[ x]^2 QPochhammer[ x^8]^2), {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^2] EllipticTheta[ 4, 0, x^4]^4 / (EllipticTheta[ 4, 0, x] 2 x^(1/2)), {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^7 / (eta(x + A)^2 * eta(x^8 + A)^2), n))};

Expansion of q^(-1/2) * eta(q^2) * eta(q^4)^7 / (eta(q)^2 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ 2, 1, 2, -6, 2, 1, 2, -4, ...].
a(n) = (-1)^n * A209940(n) = (-1)^floor(n/2) * A113419(n) = (-1)^(n + floor(n/2)) * A113417(n).

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

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A113419 Expansion of phi(x)^2 * phi(-x) * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

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A209940 Expansion of psi(x^4) * phi(-x^4)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta function.

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A258096 Expansion of psi(x^4) * phi(-x^4)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta function.

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

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