A121613 -id:A121613 - 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

A258831 Expansion of (psi(-x^3) * f(-x, x^2))^2 in powers of x where psi(), f(,) are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 3, -4, 5, -8, 7, -8, 9, -10, 14, -12, 16, -14, 15, -20, 17, -18, 19, -24, 26, -22, 23, -28, 25, -32, 32, -28, 29, -30, 38, -32, 33, -40, 40, -44, 42, -38, 39, -40, 57, -42, 43, -44, 45, -62, 47, -56, 49, -56, 62, -52, 53, -60, 64, -68, 64, -58, 59, -60

Offset: 0

Michael Somos, Jun 11 2015

sign

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

			G.f. = 1 - 2*x + 3*x^2 - 4*x^3 + 5*x^4 - 8*x^5 + 7*x^6 - 8*x^7 + 9*x^8 + ...
G.f. = q^5 - 2*q^11 + 3*q^17 - 4*q^23 + 5*q^29 - 8*q^35 + 7*q^41 - 8*q^47 + ...

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. A098098, A121613, A208435, A208457, A258832.

List([0..70], n -> (-1)^n*Sigma(6*n+5)/6); # Muniru A Asiru, Jan 30 2018

[(-1)^n*SumOfDivisors(6*n+5)/6: n in [0..70]]; // Vincenzo Librandi, Jan 30 2018

with(numtheory):
seq((-1)^(n-1)*sigma(6*n - 1)/6, n=1..10^3); # Muniru A Asiru, Jan 30 2018

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[ 1, 6 n + 5] / 6];
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^6]^2 QPochhammer[ x, -x] / QPochhammer[ x^3, -x^3])^2, {x, 0, n}];
Table[(-1)^n DivisorSigma[1, 6 n + 5] / 6, {n, 0, 60}] (* Vincenzo Librandi, Jan 30 2018 *)

{a(n) = if(n<0, 0, (-1)^n*sigma(6*n+5)/6)};

{a(n) = my(A); if(n<0, 0, A = x*O(x^n); polcoeff((eta(x + A)*eta(x^4 + A)*eta(x^6 + A)^4/(eta(x^2+A)^2*eta(x^3+A)*eta(x^12+A)))^2, n))};

Expansion of (f(-x^6)^2 * chi(x^3) / chi(x))^2 in powers of x where chi(), f() are Ramanujan theta functions.
Expansion of q^(-5/6) * (eta(q) * eta(q^4) * eta(q^6)^4 / (eta(q^2)^2 * eta(q^3) * eta(q^12)))^2 in powers of q.
Euler transform of period 12 sequence [-2, 2, 0, 0, -2, -4, -2, 0, 0, 2, -2, -4, ...].
a(n) = (-1)^n * A098098(n) = A208435(2*n + 1) = A208457(2*n + 1). 6 * a(n) = A121613(3*n + 2).
Convolution square of A258832.

A207541 Expansion of phi(q)^3 * phi(-q) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 4, 0, -16, -8, 24, 0, -32, 24, 52, 0, -48, -32, 56, 0, -96, 24, 72, 0, -80, -48, 128, 0, -96, 96, 124, 0, -160, -64, 120, 0, -128, 24, 192, 0, -192, -104, 152, 0, -224, 144, 168, 0, -176, -96, 312, 0, -192, 96, 228, 0, -288, -112, 216, 0, -288, 192, 320, 0

Offset: 0

Michael Somos, Feb 26 2012

sign

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

			1 + 4*q - 16*q^3 - 8*q^4 + 24*q^5 - 32*q^7 + 24*q^8 + 52*q^9 - 48*q^11 + ...

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. A096727, A097723, A112610, A121613, A139093, A208435.

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]^3*EllipticTheta[3, 0, -q], {q, 0, n}]; Table[A207541[n], {n, 0, 50}] (* G. C. Greubel, Dec 16 2017 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^3))^2, n))}

Expansion of phi(-q^4)^4 + 4 * q * psi(-q^2)^4 = phi(q)^3 * phi(-q) = phi(q)^2 * phi(-q^2)^2 = psi(q)^4 * chi(-q^2)^6 = phi(-q^2)^6 / phi(-q)^2 = f(q)^6 / psi(q)^2 = f(q)^4 * chi(-q^2)^2 in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.
Euler transform of period 4 sequence [ 4, -10, 4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 128 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A112610.
G.f.: Product_{k>0} (1 - x^(2*k))^14 / ((1 - x^k)^4 * (1 - x^(4*k))^6).
a(3*n + 2) = 24 * A208435(n). a(4*n + 2) = 0. a(2*n + 1) = 4 * A121613(n). a(4*n) = A096727(n). a(4*n + 1) = 4 * A112610(n). a(4*n + 3) = -16 * A097723(n). Convolution square of A139093.

A258835 Expansion of psi(x)^3 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 3, 3, 4, 7, 6, 9, 13, 9, 10, 15, 15, 13, 19, 18, 16, 30, 21, 19, 27, 21, 31, 31, 24, 25, 39, 33, 28, 48, 30, 35, 54, 33, 34, 52, 42, 45, 51, 39, 45, 55, 51, 50, 70, 45, 46, 78, 48, 54, 80, 57, 63, 78, 54, 55, 75, 84, 58, 79, 60, 61, 117, 63, 74, 87, 72, 81

Offset: 0

Michael Somos, Jun 11 2015

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

			G.f. = 1 + 3*x + 3*x^2 + 4*x^3 + 7*x^4 + 6*x^5 + 9*x^6 + 13*x^7 + 9*x^8 + ...
G.f. = q^7 + 3*q^15 + 3*q^23 + 4*q^31 + 7*q^39 + 6*q^47 + 9*q^55 + 13*q^63 + ...

Muniru A Asiru, Table of n, a(n) for n = 0..20000 (first 1000 terms from G. C. Greubel).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A010054, A121373, A121613, A222068.

sequence := List([1..10^5],n->Sigma(8*n-1)/8); # Muniru A Asiru, Dec 31 2017

with(numtheory): seq(sigma(8*n-1)/8, n=1..1000); # Muniru A Asiru, Dec 31 2017

a[ n_] := If[ n < 0, 0, DivisorSigma[ 1, 8 n + 7] / 8];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x]^3 EllipticTheta[ 2, 0, x^4] / (16 x^(7/4)), {x, 0, n}];

{a(n) = if( n<0, 0, sigma(8*n + 7) / 8)};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^8 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)), n))};

Expansion of q^(-7/8) * eta(q^2)^6 * eta(q^8)^2 / (eta(q)^3 * eta(q^4)) in powers of q.
Euler transform of period 8 sequence [ 3, -3, 3, -2, 3, -3, 3, -4, ...].
G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 + x^k)^3 * (1 + x^(2*k)) * (1 + x^(4*k))^2.
-8 * a(n) = A121613(4*n + 3). a(n) = sigma(8*n + 7) / 8.
Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/16 = 0.6168502... (A222068). - Amiram Eldar, Mar 28 2024

A208451 Expansion of phi(q) * phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 0, 16, -8, -24, 0, 32, 24, -52, 0, 48, -32, -56, 0, 96, 24, -72, 0, 80, -48, -128, 0, 96, 96, -124, 0, 160, -64, -120, 0, 128, 24, -192, 0, 192, -104, -152, 0, 224, 144, -168, 0, 176, -96, -312, 0, 192, 96, -228, 0, 288, -112, -216, 0, 288, 192, -320

Offset: 0

Michael Somos, Feb 26 2012

sign

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

			G.f. = 1 - 4*q + 16*q^3 - 8*q^4 - 24*q^5 + 32*q^7 + 24*q^8 - 52*q^9 + 48*q^11 + ...

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. A004011, A008438, A096727, A097723, A112610, A121613.

A := Basis( ModularForms( Gamma0(16), 2), 58); A[1] - 4*A[2] + 16*A[4] - 8*A[5]; /* Michael Somos, Aug 21 2014 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4])^2, {q, 0, n}]; (* Michael Somos, Aug 21 2014 *)

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

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

Expansion of phi(-q^4)^4 - 4 * q * psi(-q^2)^4 = phi(q) * phi(-q)^3 = phi(-q)^2 * phi(-q^2)^2 = phi(-q^2)^6 / phi(q)^2 = psi(-q)^4 * chi(-q^2)^6 = f(-q)^4 * chi(-q^2)^2 = f(-q)^6 / psi(-q)^2 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of (eta(q)^2 * eta(x^2) / eta(x^4))^2 in powers of q.
Euler transform of period 4 sequence [ -4, -6, -4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 5128 (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A097723.
a(4*n + 2) = 0. a(2*n + 1) = -4 * A121613(n). a(4*n) = A096727(n). a(4*n + 1) = -4 * A112610(n). a(4*n + 3) = 16 * A097723(n). a(8*n) = A004011(n). a(8*n + 4) = -8 * A008438(n).

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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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A258831 Expansion of (psi(-x^3) * f(-x, x^2))^2 in powers of x where psi(), f(,) are Ramanujan theta functions.

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A207541 Expansion of phi(q)^3 * phi(-q) in powers of q where phi() is a Ramanujan theta function.

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A258835 Expansion of psi(x)^3 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

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A208451 Expansion of phi(q) * phi(-q)^3 in powers of q where phi() is a Ramanujan theta function.

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