A133079 -id:A133079 - cp's OEIS frontend

A133089 Expansion of f(x)^3 in powers of x where f() is a Ramanujan theta function.

1, 3, 0, -5, 0, 0, -7, 0, 0, 0, 9, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, 0, -15, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, -21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Sep 09 2007

sign

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

			G.f. = 1 + 3*x - 5*x^3 - 7*x^6 + 9*x^10 + 11*x^15 - 13*x^21 - 15*x^28 + ...
G.f. = q + 3*q^9 - 5*q^25 - 7*q^49 + 9*q^81 + 11*q^121 - 13*q^169 + ...

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

Cf. A010816, A133079.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3, {x, 0, n}]; (* Michael Somos, Jun 19 2015 *)

{a(n) = if( n<0, 0, if( issquare( 8*n+1, &n), (-1)^( (n-1) \ 4) * n))};

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

Expansion of q^(-1/8) * (eta(q^2)^3 / (eta(q) * eta(q^4)))^3 in powers of q.
Euler transform of period 4 sequence [ 3, -6, 3, -3, ...].
a(n) = b(8*n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 8).
G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 64 (t/i)^(3/2) f(t) where q = exp(2 Pi i t).
a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(9*n + 1) = 3 * a(n). a(25*n + 3) = -5 * a(n).
G.f.: Sum_{k>=0} (-1)^floor(k/2) * (2*k + 1) * x^(k*(k + 1))/2.
G.f.: ( Product_{k>0} (1 - x^k) * (1 + x^k)^2 / (1 + x^(2*k)) )^3.
a(n) = -(-1)^n * A010816(n). a(3*n) = A133079(n).

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).

A204850 Expansion of f(x)^3 - 9 * x * f(x^9)^3 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 0, -5, 0, 0, -7, 0, 0, 0, -18, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, -13, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0

Offset: 0

Michael Somos, Jan 19 2012

sign

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 6*x - 5*x^3 - 7*x^6 - 18*x^10 + 11*x^15 - 13*x^21 + 30*x^28 + ...
G.f. = q - 6*q^9 - 5*q^25 - 7*q^49 - 18*q^81 + 11*q^121 - 13*q^169 + ...

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. A133079, A133089, A202394.

a[ n_] := With[ {m = Sqrt[8 n + 1]}, If[ IntegerQ@m, m (-1)^(n + Quotient[m, 6]), 0] If[ Divisible[ m, 3], 2, 1]]; (* Michael Somos, Jun 19 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^3 - 9 x QPochhammer[ -x^9]^3, {x, 0, n}]; (* Michael Somos, Jun 19 2015 *)

{a(n) = my(m); if( issquare(8*n + 1, &m), (-1)^(m \ 6 + n) * m * ((m%3 == 0) + 1), 0)};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(-x + A)^3 - 9 * x * eta(-x^9 + A)^3, n))};

Expansion of f(x^3) * a(-x) in powers of x where f() is a Ramanujan theta function and a() is a cubic AGM theta function.
a(n) = b(8*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = - (1 + (-1)^e) * 3^(e/2), b(p^e) = (1 + (-1)^e)/2 * p^(e/2) if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 5, 7 (mod 8). - Michael Somos, Jun 19 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = -576 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133079.
a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(3*n) = A133079(n). a(9*n + 1) = -6 * A133089(n). a(25*n + 3) = -5 * a(n). a(n) = (-1)^n * A202394(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).

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A133089 Expansion of f(x)^3 in powers of x where f() is a Ramanujan theta function.

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

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A204850 Expansion of f(x)^3 - 9 * x * f(x^9)^3 in powers of x where f() is a Ramanujan theta 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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