A080332 -id:A080332 - cp's OEIS frontend

A113277 Expansion of q^(-1/3) * eta(q^2)^5 / eta(q)^2 in powers of q.

1, 2, 0, 0, 0, -4, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -17, 0, 0, 0, 0

Offset: 0

Michael Somos, Oct 21 2005

sign

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

			G.f. = 1 + 2*x - 4*x^5 - 5*x^8 + 7*x^16 + 8*x^21 - 10*x^33 - 11*x^40 + 13*x^56 + ...
G.f. = q + 2*q^4 - 4*q^16 - 5*q^25 + 7*q^49 + 8*q^64 - 10*q^100 - 11*q^121 +...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Y. Martin and K. Ono, Eta-quotients and elliptic curves, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3169-3176. MR1401749 (97m:11057)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A080332, A114855.

QP = QPochhammer; s = QP[q^2]^5/QP[q]^2 + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

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

{a(n) = if(issquare( 3*n + 1, &n), n * (-1)^(n%3 + n), 0)};

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

a(n) = b(3*n + 1) where b(n) is multiplicative and a(p^e) = 0 if e is odd, a(3^e) = 0^e, a(2^e) = -(-2)^(e/2), a(p^e) = p^(e/2) if p == 1 (mod 3), a(p^e) = (-p)^(e/2) if p == 2 (mod 3).
Euler transform of period 2 sequence [ 2, -3, ...].
G.f.: Sum_{k} (3*k + 1) * (-x)^(3*k^2 + 2*k) = Product_{k>0} (1 - x^k)^3 * (1 + x^k)^5.
Expansion of psi(x^2) * f(x)^2 = phi(x) * f(-x^4)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3456^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A080332.
a(4*n + 2) = a(4*n + 3) = a(5*n + 2) = a(5*n + 4) = a(8*n + 4) = 0. a(25*n + 8) = -5 * a(n). A114855(n) = (-1)^n * a(n). a(4*n + 1) = 2 * A114855(n). a(8*n) = A080332(n).
G.f.: Product_{n >= 1} (1 - q^(4*n))^3 * (1 + q^(4*n-1))^2 * (1 - q^(4*n-2)) * (1 + q^(4*n-3))^2 = Product_{n >= 1} (1 - q^(4*n))^3 * (1 - q^(4*n-1))^(-2) * (1 - q^(4*n-2))^3 * (1 - q^(4*n-3))^(-2). - Peter Bala, Jun 07 2025

A360191 G.f. 1 / Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2.

Original entry on oeis.org

1, 5, 18, 55, 149, 371, 867, 1923, 4086, 8374, 16634, 32152, 60669, 112041, 202943, 361200, 632647, 1091917, 1859225, 3126242, 5195715, 8541624, 13899866, 22404091, 35787815, 56683294, 89061028, 138872410, 214984454, 330532633, 504869316, 766357010, 1156355165

Offset: 0

Paul D. Hanna, Jan 29 2023

nonn

Self-convolution inverse of A080332.

			G.f.: A(x) = 1 + 5*x + 18*x^2 + 55*x^3 + 149*x^4 + 371*x^5 + 867*x^6 + 1923*x^7 + 4086*x^8 + 8374*x^9 + 16634*x^10 + 32152*x^11 + 60669*x^12 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A080332, A361535, A361050, A361550.

nmax = 30; CoefficientList[Series[1/Product[(1 - x^k)^3 * (1 - x^(2*k-1))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 07 2023 *)
nmax = 30; CoefficientList[Series[1/(QPochhammer[x] * EllipticTheta[4, 0, x]^2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 07 2023 *)

{a(n) = polcoeff( 1/prod(m=1,n, (1 - x^m)^3 * (1 - x^(2*m-1))^2 +x*O(x^n)), n)}
for(n=0,32,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(!) A(x) = 1 / [Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2].
(2) A(x) = 1 / [Sum_{n=-oo..+oo} (6*n + 1) * x^(n*(3*n + 1)/2)].
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (12*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Feb 07 2023

A114855 Expansion of q^(-1/3) * (eta(q) * eta(q^4))^2 / eta(q^2) in powers of q.

Original entry on oeis.org

1, -2, 0, 0, 0, 4, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -17, 0, 0, 0, 0

Offset: 0

Michael Somos, Jan 01 2006

sign

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

			G.f. = 1 - 2*x + 4*x^5 - 5*x^8 + 7*x^16 - 8*x^21 + 10*x^33 - 11*x^40 + ...
G.f. = q - 2*q^4 + 4*q^16 - 5*q^25 + 7*q^49 - 8*q^64 + 10*q^100 - 11*q^121 + ...

S. Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 147, Ed. G. H. Hardy et al., AMS Chelsea 2000.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266. MR0099904 (20 #6340)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
M. Josuat-Verges and J. S. Kim, Touchard-Riordan formulas, T-fractions, and Jacobi's triple product identity, p. 28, equation (61), arXiv:1101.5608, 2011

Cf. A080332, A113277, A114855.

A := Basis( CuspForms( Gamma1(36), 3/2), 300); A[1] - 2*A[4]; /* Michael Somos, Mar 11 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x^4]^2 EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Mar 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}];  (* Michael Somos, Mar 11 2015 *)
a[ n_] := With[{m = Sqrt[3 n + 1]}, If[ IntegerQ[ m], -m (-1)^Mod[ m, 3], 0]]; (* Michael Somos, Mar 11 2015 *)

{a(n) = if( issquare( 3*n + 1, &n), n * -(-1)^(n%3), 0)};

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

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

Expansion of psi(x^2) * f(-x)^2 = phi(-x) * f(-x^4)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -2, -1, -2, -3, ...].
a(n) = b(3*n + 1) where b(n) is multiplicative and a(p^e) = 0 if e is odd, a(3^e) = 0^e, a(p^e) = p^(e/2) if p == 1 (mod 3), a(p^e) = (-p)^(e/2) if p == 2 (mod 3).
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = (u*w * (u + 2*w) * (u + 4*w))^2 - v^6 * (u^2 + 4*u*w + 8*w^2).
G.f.: Sum_{k} (3*k + 1) * x^(3*k^2 + 2*k) = Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)) * (1 - x^(4*k)).
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0. a(4*n + 1) = -2 * a(n). 2 * a(n) = A113277(4*n + 1) = - A114855(4*n + 1).
(-1)^n * a(n) = A113277(n). a(8*n) = A080332(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6^(3/2) (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - Michael Somos, Mar 11 2015

A134756 Coefficients of a q-series of Zagier related to the Dedekind eta function.

Original entry on oeis.org

1, -5, -7, 0, 0, 11, 0, 13, 0, 0, 0, 0, -17, 0, 0, -19, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, -29, 0, 0, 0, 0, -31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -41, 0, 0, 0, 0, 0, 0, -43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 47, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Nov 08 2007

sign

Obtained by formally "differentiating the Dedekind eta-function half a time".

			G.f. = 1 - 5*x - 7*x^2 + 11*x^5 + 13*x^7 - 17*x^12 - 19*x^15 + 23*x^22 + ...
G.f. = q - 5*q^25 - 7*q^49 + 11*q^121 + 13*q^169 - 17*q^289 - 19*q^361 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Don Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function, Topology, vol.40, pp.945-960 (2001). See Eq. (20).

Cf. A010815.

Apart from signs, same as A080332, A116916 and A133079. - N. J. A. Sloane, Nov 11 2007

a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ @ m, m KroneckerSymbol[ 12, m], 0]]; (* Michael Somos, Oct 15 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (If[ # < 5, 0, (1 - Mod[#2, 2]) (# KroneckerSymbol[ 12, #])^(#2/2)] & @@@ FactorInteger[ 24 n + 1])]; (* Michael Somos, Oct 15 2015 *)
s = QPochhammer[q] + O[q]^100; A010815 = CoefficientList[s, q]; nn = Range[0, Length[A010815]-1]; A134756 = Sqrt[24*nn+1]*A010815 (* Jean-François Alcover, Dec 01 2015 *)

{a(n) = if( issquare( 24*n+1, &n), n * kronecker( 12, n), 0)};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(24*n+1); prod(k = 1, matsize(A)[1], [p, e] = A[k, ]; if( (p<5) || (e%2), 0, (kronecker( 12, p) * p)^(e\2))))};

a(n) = b(24*n + 1) where b() is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 11 (mod 12), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 12).
G.f.: Sum_{k>0} Kronecker(12, k) * k * x^((k^2 - 1) / 24).
a(n) = sqrt(24*n + 1) * A010815(n).

A178902 Expansion of q^(-1/24) * eta(q^2)^13 / (eta(q)^5 * eta(q^4)^5) in powers of q.

Original entry on oeis.org

1, 5, 7, 0, 0, 11, 0, -13, 0, 0, 0, 0, -17, 0, 0, -19, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, -37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -41, 0, 0, 0, 0, 0, 0, -43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Jun 21 2010

sign

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

			G.f. = 1 + 5*x + 7*x^2 + 11*x^5 - 13*x^7 - 17*x^12 - 19*x^15 - 23*x^22 + ...
G.f. = q + 5*q^25 + 7*q^49 + 11*q^121 - 13*q^169 - 17*q^289 - 19*q^361 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
G. Köhler, Some eta-identities arising from theta series, Math. Scand. 66 (1990), 147-154.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Apart from signs, same as A080332, A116916, A133079 and A134756.

A178902[n_] := SeriesCoefficient[(QPochhammer[-q, -q]/QPochhammer[q, -q])^3/QPochhammer[-q, q^2], {q, 0, n}]; Table[A178902[n], {n, 0, 50}] (* G. C. Greubel, Aug 17 2017 *)
a[ n_] := With[ {m = Sqrt[24 n + 1]}, If[ IntegerQ@m, m KroneckerSymbol[ -6, m], 0]]; (* Michael Somos, Apr 27 2018 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^13 / (QPochhammer[ x] QPochhammer[ x^4])^5, {x, 0, n}]; (* Michael Somos, Apr 27 2018 *)

{a(n) = if( issquare( 24*n + 1, &n), n * kronecker( -6, n), 0)};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(24*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( (p<5) || (e%2), 0, if( p%24<12, p, -p)^(e\2))))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^13 / (eta(x + A)^5 * eta(x^4 + A)^5), n))};

Expansion of f(q) * phi(q)^2 = f(q)^3 * chi(q)^2 = phi(q)^3 / chi(q) in powers of q where f(), phi(), chi() are Ramanujan theta functions.
Euler transform of period 4 sequence [5, -8, 5, -3, ...].
a(n) = b(24*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 5, 7, 11 (mod 24), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 13, 17, 19, 23 (mod 24).
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 48^(3/2) (t/i)^(3/2) f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^(2*k))^3 * (1 + x^(2*k - 1))^5 = Sum_{k>0} Kronecker( -6, k) * k * x^((k^2 - 1) / 24) = Sum_{k in Z} (6*k + 1) * (-1)^floor(k/2) * x^(k * (3*k + 1) / 2).
a(n) = (-1)^n * A080332(n).

cp's OEIS Frontend

A113277 Expansion of q^(-1/3) * eta(q^2)^5 / eta(q)^2 in powers of q.

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A360191 G.f. 1 / Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2.

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A114855 Expansion of q^(-1/3) * (eta(q) * eta(q^4))^2 / eta(q^2) in powers of q.

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A134756 Coefficients of a q-series of Zagier related to the Dedekind eta function.

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A178902 Expansion of q^(-1/24) * eta(q^2)^13 / (eta(q)^5 * eta(q^4)^5) in powers of q.

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