A115978 -id:A115978 - cp's OEIS frontend

A115979 Expansion of (1 - theta_4(q)*theta_4(q^3))/2 in powers of q.

1, 0, 1, -3, 0, 0, 2, 0, 1, 0, 0, -3, 2, 0, 0, -3, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, -6, 0, 0, 2, 0, 0, 0, 0, -3, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, -3, 3, 0, 0, -6, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, -3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, -6, 0, 0, 2, 0, 1, 0, 0, -6, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 0, -3, 0, 0, 2, 0, 0

Offset: 1

Michael Somos, Feb 09 2006

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A033716, A096936, A115978.

S:=series((1-JacobiTheta4(0,q)*JacobiTheta4(0,q^3))/2, q, 106):
seq(coeff(S,q,n),n=1..105); # Robert Israel, Nov 20 2017

Drop[CoefficientList[Series[(1 -EllipticTheta[4, 0, q]*EllipticTheta[4, 0, q^3])/2, {q, 0, 110}], q], 1] (* G. C. Greubel, May 09 2019 *)

{a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( (eta(x+A)*eta(x^3+A))^2/eta(x^2+A)/eta(x^6+A), n)/-2)}

def E(x): return 1 + 2*sum((-1)^k*x^(k^2) for k in (1..50))
a=((1 - E(x)*E(x^3))/2).series(x, 110).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 09 2019

(define (A115979 n) (- (* (expt -1 n) (A096936 n)))) ;; Follow A096936 for the rest of code. - Antti Karttunen, Nov 20 2017

Expansion of (1-(eta(q)*eta(q^3))^2/(eta(q^2)*eta(q^6)))/2 in powers of q.
Moebius transform is period 12 sequence [1,-1,0,-3,-1,0,1,3,0,1,-1,0,...].
a(n) is multiplicative and a(2^e) = -3(1+(-1)^e)/2 if e>0, a(3^e)=1, a(p^e) = 1+e if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} x^(k)/(1+x^k+x^(2k)) -4x^(4k)/(1+x^(4k)+x^(8k)).
a(n) = -(-1)^n*A096936(n).
A115978(n) = -2*a(n) if n > 0.

A097109 G.f.: s(2)^2s(3)^3/(s(1)s(6)^2), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.

Original entry on oeis.org

1, 1, 0, -2, -3, 0, 0, 2, 0, -2, 0, 0, 6, 2, 0, 0, -3, 0, 0, 2, 0, -4, 0, 0, 0, 1, 0, -2, -6, 0, 0, 2, 0, 0, 0, 0, 6, 2, 0, -4, 0, 0, 0, 2, 0, 0, 0, 0, 6, 3, 0, 0, -6, 0, 0, 0, 0, -4, 0, 0, 0, 2, 0, -4, -3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, -2, -6, 0, 0, 2, 0, -2, 0, 0, 12, 0, 0, 0, 0, 0, 0, 4, 0, -4, 0, 0, 0, 2, 0, 0, -3, 0, 0, 2, 0

Offset: 0

N. J. A. Sloane, Sep 16 2004

Coefficients are multiplicative [Fine].

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.36).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A010815, A115978, A122861, A248897.

QP = QPochhammer; s = QP[q^2]^2*(QP[q^3]^3/(QP[q]*QP[q^6]^2)) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, adapted from PARI *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 / (eta(x + A) * eta(x^6 + A)^2), n))} /* Michael Somos, Sep 15 2006 */

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if(p==2, 3*(e%2-1), if( p==3, -2, if( p%6==1, e+1, 1-e%2))))))} /* Michael Somos, Sep 15 2006 */

Fine gives an explicit formula for a(n) in terms of the divisors of n.
From Michael Somos, Sep 15 2006: (Start)
Expansion of (a(q) - 3*a(q^3) - 4*a(q^4) + 12*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.
Euler transform of period 6 sequence [ 1, -1, -2, -1, 1, -2, ...].
a(n) is multiplicative with a(2^e) = -3(1+(-1)^e)/2 if e>0, a(3^e) = -2 if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6). (End)
a(3*n + 2) = 0. a(3*n) = A115978(n). a(3*n + 1) = A122861(n).
Sum_{k=0..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 22 2024

A164272 Expansion of phi(q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 11 2009

sign

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

			G.f. = 1 + 2*q - 2*q^3 - 2*q^4 - 4*q^7 + 2*q^9 - 2*q^12 + 4*q^13 + 6*q^16 + ...

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. A112604, A112605, A115978, A129449, A164273, A246752.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^3], {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)
f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164272[n_] := SeriesCoefficient[f[q, q]*f[-q^3, -q^3], {q, 0, n}]; Table[A164272[n], {n,0,50}] (* G. C. Greubel, Sep 16 2017 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)), n))};

Expansion of eta(q^2)^5 * eta(q^3)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, -3, 0, -1, 2, -4, 2, -1, 0, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 768^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112604.
a(n) = (-1)^n * A164273(n).
a(2*n + 1) = 2 * A129449(n). a(4*n) = A115978(n). a(4*n + 1) = 2 * A112604(n). a(4*n + 2) = 0. a(4*n + 3) = -2 * A112605(n).
a(3*n) = A164273(n). a(3*n + 1) = 2 * A246752(n). a(3*n + 2) = 0. - Michael Somos, Sep 02 2015

A217221 Theta series of Kagome net with respect to a deep hole.

Original entry on oeis.org

0, 6, 0, 6, 0, 0, 0, 12, 0, 6, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 12, 0, 0, 0, 6, 0, 6, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 12, 0, 0, 0, 12, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 12, 0, 12, 0, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 6, 0, 0, 0, 12, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24, 0, 12, 0, 0, 0, 12, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Oct 05 2012

nonn

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

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

			G.f. = 6*q + 6*q^3 + 12*q^7 + 6*q^9 + 12*q^13 + 12*q^19 + 12*q^21 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.

Antti Karttunen, Table of n, a(n) for n = 0..65537
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A033762, A115978, A217220.

A := Basis( ModularForms( Gamma1(12), 1), 80); 6*A[2] + 6*A[4]; /* Michael Somos, Feb 01 2017 */

a[ n_] := If[ n < 1 || EvenQ[n], 0, 6 DivisorSum[n, Mod[(3 - #)/2, 3, -1] &]]; (* Michael Somos, Feb 01 2017 *)

{a(n) = if( n<1 || n%2==0, 0, 6 * sumdiv(n, d, kronecker(-3, d)))}; /* Michael Somos, Feb 01 2017 */

Phi_0(q)-phi_0(q^4) in the notation of SPLAG, Chapter 4.
Expansion of a(q) - a(q^4) in powers of q where a() is a cubic AGM function. - Michael Somos, Feb 01 2017
Expansion of 6 * q * psi(q^2) * psi(q^6) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 01 2017
Expansion of 6 * (eta(q^4) * eta(q^12))^2 / (eta(q^2) * eta(q^6)) in powers of q. - Michael Somos, Feb 01 2017
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 27^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A115978. - Michael Somos, Feb 01 2017
a(2*n) = 0. a(2*n + 1) = 6 * A033762(n). - Michael Somos, Feb 01 2017

cp's OEIS Frontend

A115979 Expansion of (1 - theta_4(q)*theta_4(q^3))/2 in powers of q.

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A097109 G.f.: s(2)^2*s(3)^3/(s(1)*s(6)^2), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.

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A164272 Expansion of phi(q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

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A217221 Theta series of Kagome net with respect to a deep hole.

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Magma

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A097109 G.f.: s(2)^2s(3)^3/(s(1)s(6)^2), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.