A129134 -id:A129134 - cp's OEIS frontend

A082564 Expansion of eta(q)^2 * eta(q^2) / eta(q^4) in powers of q.

1, -2, -2, 4, 2, 0, -4, 0, 2, -6, 0, 4, 4, 0, 0, 0, 2, -4, -6, 4, 0, 0, -4, 0, 4, -2, 0, 8, 0, 0, 0, 0, 2, -8, -4, 0, 6, 0, -4, 0, 0, -4, 0, 4, 4, 0, 0, 0, 4, -2, -2, 8, 0, 0, -8, 0, 0, -8, 0, 4, 0, 0, 0, 0, 2, 0, -8, 4, 4, 0, 0, 0, 6, -4, 0, 4, 4, 0, 0, 0, 0, -10, -4, 4, 0, 0, -4, 0, 4, -4, 0, 0, 0, 0, 0, 0, 4, -4, -2, 12, 2, 0, -8, 0

Offset: 0

Benoit Cloitre, May 05 2003

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is nonzero if and only if n is in A002479. - Michael Somos, Dec 15 2011
Absolute values appear to give A033715 = 2*A002325.
Denoted by a_4(n) in Kassel and Reutenauer 2015. - Michael Somos, Jun 04 2015

			G.f. = 1 - 2*q - 2*q^2 + 4*q^3 + 2*q^4 - 4*q^6 + 2*q^8 - 6*q^9 + 4*q^11 + 4*q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015 , see page 31 7.2(c). [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016, see p. 13 paragraph 3.3.3.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002479, A033715, A033761, A125095, A129134, A133692, A258747, A258764.

A := Basis( ModularForms( Gamma1(32), 1), 105); A[1] - 2*A[2] - 2*A[3] + 4*A[4] + 2*A[5] - 4*A[7] + 2*A[9] - 6*A[10] + 4*A[12] + 4*A[13] - 4*A[16]; /* Michael Somos, Aug 29 2014 */

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^2] / QPochhammer[ q^4], {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 2 (-1)^Quotient[ n + 1, 2] DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* Michael Somos, Aug 29 2014 *)

{a(n) = if( n<1, n==0, 2 * (-1)^((n+1) \ 2) * sumdiv( n, d, kronecker( -2, d)))}; /* Michael Somos, Mar 30 2007 */

{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), n))};

Expansion of phi(-q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 30 2007
Euler transform of period 4 sequence [ -2, -3, -2, -2, ...]. - Michael Somos, Mar 30 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(11/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033761. - Michael Somos, Aug 29 2014
G.f.: Product_{k>0} (1 - x^k)^2 / (1 + x^(2*k)). - Michael Somos, Mar 30 2007
a(n) = -2 * A129134(n) unless n=0. - Michael Somos, Mar 30 2007
a(n) = (-1)^floor( (n+1)/2 ) * A033715(n). - Michael Somos, Aug 29 2014
a(2*n) = A133692(n). a(2*n + 1) = -2 * A125095(n). - Michael Somos, Aug 29 2014
a(3*n + 1) = -2 * A258747(n). a(3*n + 2) = -2 * A258764(n). - Michael Somos, Jun 09 2015

A258747 Expansion of chi(-x) * f(x^3) * f(-x^6) in powers of x where chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

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 - x - x^5 - 2*x^6 + 2*x^7 + x^8 + 2*x^11 - 2*x^14 + x^16 - x^21 + ...
G.f. = q - q^4 - q^16 - 2*q^19 + 2*q^22 + q^25 + 2*q^34 - 2*q^43 + q^49 + ...

G. C. Greubel, 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. A082564, A129134, A255318, A255319, A257398.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3] QPochhammer[ x^6], {x, 0, n}];
a[ n_] := If[ n < 0, 0, (-1)^Quotient[ 3 n, 2] DivisorSum[ 3 n + 1, KroneckerSymbol[-2, #] &]];

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

{a(n) = if( n<0, 0, (-1)^(3*n\2) * sumdiv(3*n + 1, d, kronecker( -2, d)))};

Expansion of q^(-1/3) * eta(q) * eta(q^6)^4 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 0, 0, -1, -3, -1, 0, 0, 0, -1, -2, ...].
G.f.: Product_{k>0} (1 + x^(3*k)) * (1 - x^(6*k))^2 / ( (1 + x^k) * (1 + x^(6*k)) ).
-2 * a(n) = A082564(3*n + 1). a(n) = A129134(3*n + 1).
a(4*n + 3) = 2 * A257402(n-1). a(8*n) = A257398(n). a(8*n + 2) = a(8*n + 4) = a(16*n + 3) = a(16*n + 15) = 0. a(16*n + 7) = 2 * A255318(n). a(16*n + 11) = 2 * A255319(n).

A258764 Expansion of chi(-x^2) * psi(-x^3)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

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 - x^2 - 2*x^3 + 2*x^5 - x^10 + 2*x^12 + 2*x^13 - 2*x^14 + x^16 + ...
G.f. = q^2 - q^8 - 2*q^11 + 2*q^17 - q^32 + 2*q^38 + 2*q^41 - 2*q^44 + ...

G. C. Greubel, 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. A082564, A129134, A255317, A255318, A255319, A257399, A257402.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^4] QPochhammer[ x^3]^2 / QPochhammer[ x^6, x^12]^2, {x, 0, n}];
a[ n_] := If[ n < 0, 0, (-1)^Quotient[ n, 2] DivisorSum[ 3 n + 2, KroneckerSymbol[-2, #] &]];

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

{a(n) = if( n<0, 0, (-1)^(n\2) * sumdiv(3*n + 2, d, kronecker( -2, d)))};

Expansion of q^(-2/3) * eta(q^2) * eta(q^3)^2 * eta(q^12)^2 / (eta(q^4) * eta(q^6)^2) in powers of q.
Euler transform of period 12 sequence [ 0, -1, -2, 0, 0, -1, 0, 0, -2, -1, 0, -2, ...].
G.f.: Product_{k>0} (1 + x^(2*k)) * (1 - x^(3*k))^2 * (1 - x^(2*k) + x^(4*k))^2.
a(n) = A129134(3*n + 2). -2 * a(n) = A082564(3*n + 2).
a(4*n) = A257399(n). a(8*n + 3) = -2 * A255318(n). a(8*n + 5) = 2 * A255319(n). a(8*n + 6) = -2 * A257402(n-1). a(16*n) = A257398(n). a(16*n + 2) = - A257399(n). a(16*n + 12) = 2 * A255317(n).
a(8*n + 1) = a(8*n + 7) = a(16*n + 4) = a(16*n + 8) = 0.

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A082564 Expansion of eta(q)^2 * eta(q^2) / eta(q^4) in powers of q.

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A258747 Expansion of chi(-x) * f(x^3) * f(-x^6) in powers of x where chi(), f() are Ramanujan theta functions.

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A258764 Expansion of chi(-x^2) * psi(-x^3)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

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