A209942 -id:A209942 - cp's OEIS frontend

A134343 Expansion of psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.

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

Offset: 0

Michael Somos, Oct 21 2007

sign

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

Number 57 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 2*x + x^2 - 2*x^3 + 2*x^4 + 3*x^6 - 2*x^7 - 2*x^9 + 2*x^10 + ...
G.f. = q - 2*q^5 + q^9 - 2*q^13 + 2*q^17 + 3*q^25 - 2*q^29 - 2*q^37 + ...

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. A008441, A053692, A113407, A121613, A143379, A209942, A213791, A246862, A246683, A246950, A259285, A259287.

A := Basis( ModularForms( Gamma1(64), 1), 321); A[2] - 2*A[6] + A[10] - 2*A[14] + 2*A[18] + 3*A[26] - 2*A[30] + 2*A[35] - 2*A[36]; /* Michael Somos, Jun 22 2015 */;

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 4 n + 1, (-1)^Quotient[#, 2] &]]; (* Michael Somos, Jun 22 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(1/2)]^2 / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, Jun 22 2015 *)
a[ n_] := SeriesCoefficient[(QPochhammer[ x] QPochhammer[ x^4] / QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Jun 22 2015 *)

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

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

Expansion of q^(-1/4) * (eta(q) * eta(q^4) / eta(q^2))^2 in powers of q.
Euler transform of period 4 sequence [ -2, 0, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 8 (t/i) f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 8), b(p^e) = (-1)^e * (e+1) if p == 5 (mod 8).
G.f.: (Product_{k>0} (1 - x^k) * (1 + x^(2*k)))^2.
a(9*n + 5) = a(9*n + 8) = 0. a(n) = (-1)^n * A008441(n). a(2*n) = A113407(n). a(2*n + 1) = -2 * A053692(n).
2 * a(n) = A204531(4*n + 1) = - A246950(n). a(4*n) = A246862(n). a(4*n + 2) = A246683(n). - Michael Somos, Jun 22 2015
a(4*n + 1) = -2 * A259287(n). a(4*n + 3) = -2 * A259285(n). - Michael Somos, Jun 24 2015
Convolution square is A121613. Convolution cube is A213791. Convolution with A000009 is A143379. Convolution with A000143 is A209942. Michael Somos, Jun 22 2015
G.f.: Sum_{k>0 odd} (x^k + x^(3*k)) / (1 + x^(4*k)) * (-1)^floor((k+1) / 4). - Michael Somos, Jun 22 2015
G.f.: Sum_{k>0 odd} (x^k - x^(3*k)) / (1 + x^(4*k)) * (-1)^floor(k / 4). - Michael Somos, Jun 22 2015

A215472 Expansion of (psi(x) * phi(-x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -14, 81, -238, 322, 0, -429, 82, 0, 2162, -3038, -1134, 2401, 2482, 0, -6958, 3332, 0, 1442, 0, 6561, -4508, -9758, 0, -1918, 18802, 0, 9362, -24638, -19278, 14641, 14756, 0, 0, 6562, 0, -1148, -33998, 26082, -20398, 0, 0, 28083, 49042, 0, -64078, -30268

Offset: 0

Michael Somos, Aug 12 2012

sign

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

This is a member of an infinite family of integer weight level 8 modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472.

			1 - 14*x + 81*x^2 - 238*x^3 + 322*x^4 - 429*x^6 + 82*x^7 + 2162*x^9 + ...
q - 14*q^5 + 81*q^9 - 238*q^13 + 322*q^17 - 429*q^25 + 82*q^29 + 2162*q^37 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Shobhit, How to prove that eta(q^4)^14/eta(q^8)^4 = 4eta(q^2)^4eta(q^4)^2eta(q^8)^4 + eta(q)^4eta(q^2)^2eta(q^4)^4?
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000729, A002171, A008441, A030212, A209942, A215601.

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^14 / QPochhammer[ x^2]^4, {x, 0, n}] (* Michael Somos, Sep 05 2013 *)

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

Expansion of q^(-1/4) * eta(q)^14 / eta(q^2)^4 in powers of q.
Expansion of q^(-1/4) * ( eta(q)^4 * eta(q^2)^2 * eta(q^4)^4 + 4 * eta(q^2)^4 * eta(q^4)^2 * eta(q^8)^4 ) in powers of q. - Michael Somos, Sep 05 2013
Euler transform of period 2 sequence [ -14, -10, ...].
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 128 (t/i)^5 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A030212.
a(n) = (-1)^n * A209942(n). a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = 81 * a(n).
a(n) = A030212(4*n + 1). - Michael Somos, Sep 05 2013

A258771 Expansion of psi(-x) * phi(x)^4 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 7, 16, 7, -16, 0, 17, -48, -64, 16, 1, -16, 16, -32, 32, 55, -48, 64, 64, 16, 128, -9, -80, -32, 16, 48, -80, 96, 49, -144, -16, -144, -64, -64, -96, 144, 33, -64, -160, 0, 112, 32, 32, -96, 128, -25, 0, 32, -160, 304, 144, 96, 144, -48, 48, 119, 16, -256

Offset: 0

Michael Somos, Jun 09 2015

sign

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

			G.f. = 1 + 7*x + 16*x^2 + 7*x^3 - 16*x^4 + 17*x^6 - 48*x^7 - 64*x^8 + ...
G.f. = q + 7*q^9 + 16*q^17 + 7*q^25 - 16*q^33 + 17*q^49 - 48*q^57 + ...

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. A209942, A258770.

a[ n_] := SeriesCoefficient[EllipticTheta[ 3, 0, x]^4 QPochhammer[ x] / QPochhammer[ x^2, x^4], {x, 0, n}];

{a(n) = my(A);  if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^19 / (eta(x + A) * eta(x^4 + A) )^7, n))};

Expansion of q^(-1/8) * eta(q^2)^19 / (eta(q) * eta(q^4))^7 in powers of q.
Euler transform of period 4 sequence [ 7, -12, 7, -5, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 1024 (t/i)^(5/2) f(t) where q = exp(2 Pi i t).
a(3*n + 2) = 16 * A258770(n).
Convolution square is A209942.

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A134343 Expansion of psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.

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A215472 Expansion of (psi(x) * phi(-x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

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A258771 Expansion of psi(-x) * phi(x)^4 in powers of x where phi(), psi() are Ramanujan theta functions.

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