A097195 -id:A097195 - cp's OEIS frontend

A227696 Expansion of f(x^3)^3 / f(x) in powers of x where f() is a Ramanujan theta function.

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

Offset: 0

Michael Somos, Sep 22 2013

sign

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

			G.f. = 1 - x + 2*x^2 + 2*x^4 - x^5 + 2*x^6 + x^8 - 2*x^9 + 2*x^10 + ...
G.f. = q - q^4 + 2*q^7 + 2*q^13 - q^16 + 2*q^19 + q^25 - 2*q^28 + 2*q^31 + ...

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. A033687, A097195, A121361, A123884, A226535.

a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3]^3 / QPochhammer[ -q], {q, 0, n}]

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^9 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^3, n))}

Expansion of q^(-1/3) * eta(q) * eta(q^4) * eta(q^6)^9 / (eta(q^2) * eta(q^3) * eta(q^12))^3 in powers of q.
Euler transform of period 12 sequence [ -1, 2, 2, 1, -1, -4, -1, 1, 2, 2, -1, -2, ...].
Moebius transform is period 36 sequence [ 1, -1, -1, -1, -1, 1, 1, 1, 0, 1, -1, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 1, -1, 0, -1, -1, -1, 1, 1, 1, 1, -1, 0, ...].
a(n) = b(3*n+1) where b(n) is multiplicative with b(2^e) = - (1 + (-1)^e) / 2 if e>0, b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^3) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (4/3)^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A226535.
G.f.: Product_{k>0} (1 - (-x)^k)^3 / (1 - (-x)^k).
a(n) = (-1)^n * A033687(n). a(4*n + 3) = 0.
a(2*n) = A097195(n). a(4*n) = A123884(n). a(4*n + 1) = - A033687(n). a(4*n + 2) = 2 * A121361(n).

A378007 Square table read by descending antidiagonals: T(n,k) = A378006(k*n+1,k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 0, 1, 1, 1, 2, 4, 2, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 0, 2, 0, 0, 2, 1, 1, 1, 0, 4, 0, 1, 0, 3, 0, 1, 1, 1, 4, 6, 2, 6, 2, 4, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 1, 1, 4, 10, 4, 6, 4, 6, 2, 4, 2, 2, 1, 1

Offset: 0

Jianing Song, Nov 14 2024

A condensed version of A378006: the k-th column is the sequence {b(k*n+1)}, with the sequence {b(n)} having Dirichlet g.f. Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo k.

			Table starts
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
  1, 1, 1, 2, 0, 2, 2, 2, 0, 4, ...
  1, 1, 2, 1, 4, 2, 0, 4, 6, 0, ...
  1, 1, 0, 2, 1, 2, 0, 2, 0, 4, ...
  1, 1, 2, 2, 0, 1, 6, 0, 6, 4, ...
  1, 1, 1, 0, 0, 2, 0, 4, 0, 0, ...
  1, 1, 2, 3, 4, 2, 6, 2, 0, 4, ...
  1, 1, 0, 2, 0, 2, 0, 0, 1, 4, ...
  1, 1, 1, 0, 4, 3, 0, 0, 6, 1, ...
  1, 1, 2, 2, 0, 0, 3, 4, 0, 0, ...
  1, 1, 2, 2, 0, 2, 6, 3, 0, 4, ...
Write w = exp(2*Pi*i/3) = (-1 + sqrt(3)*i)/2.
Column k = 1: 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + 1/7^s + 1/8^s + 1/9^s + 1/10^s + 1/11^s + ...;
Column k = 2: 1 + 1/3^s + 1/5^s + 1/7^s + 1/9^s + 1/11^s + 1/13^s + 1/15^s + 1/17^s + 1/19^s + 1/21^s + ...;
Column k = 3: (1 + 1/2^s + 1/4^s + 1/5^s + ...)*(1 - 1/2^s + 1/4^s - 1/5^s + ...) = 1 + 1/4^s + 2/7^s + 2/13^s + 1/16^s + 2/19^s + 1/25^s + 2/28^s + 2/31^s + ...;
Column k = 4: (1 + 1/3^s + 1/5^s + 1/7^s + ...)*(1 - 1/3^s + 1/5^s - 1/7^s + ...) = 1 + 2/5^s + 1/9^s + 2/13^s + 2/17^s + 3/25^s + 2/29^s + 2/37^s + 2/41^s + ...;
Column k = 5: (1 + 1/2^s + 1/3^s + 1/4^s + ...)*(1 + i/2^s - i/3^s - 1/4^s + ...)*(1 - 1/2^s - 1/3^s + 1/4^s + ...)*(1 - i/2^s + i/3^s - 1/4^s + ...) = 1 + 4/11^s + 1/16^s + 4/31^s + 4/41^s + ...;
Column k = 6: (1 + 1/5^s + 1/7^s + 1/11^s + ...)*(1 - 1/5^s + 1/7^s - 1/11^s + ...) = 1 + 2/7^s + 2/13^s + 2/19^s + 1/25^s + 1/31^s + 2/37^s + 2/43^s + 3/49^s + 2/61^s + ...;
Column k = 7: (1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + ...)*(1 + w/2^s + (w+1)/3^s - (w+1)/4^s - w/5^s - 1/6^s + ...)*(1 - (w+1)/2^s + w/3^s + w/4^s - (w+1)/5^s + 1/6^s + ...)*(1 + 1/2^s - 1/3^s + 1/4^s - 1/5^s - 1/6^s + ...)*(1 + w/2^s - (w+1)/3^s - (w+1)/4^s + w/5^s + 1/6^s + ...)*(1 - (w+1)/2^s - w/3^s + w/4^s + (w+1)/5^s - 1/6^s + ...) = 1 + 2/8^s + 6/29^s + 6/43^s + 3/64^s + 6/71^s + ...;
Column k = 8: (1 + 1/3^s + 1/5^s + 1/7^s + ...)*(1 + 1/3^s - 1/5^s - 1/7^s + ...)*(1 - 1/3^s + 1/5^s - 1/7^s + ...)*(1 - 1/3^s - 1/5^s + 1/7^s + ...) = 1 + 2/9^s + 4/17^s + 2/25^s + 4/41^s + 2/49^s + 4/73^s + 3/81^s + ...;
Column k = 9: (1 + 1/2^s + 1/4^s + 1/5^s + 1/7^s + 1/8^s + ...)*(1 + (w+1)/2^s + w/4^s - w/5^s - (w+1)/7^s - 1/8^s + ...)*(1 + w/2^s - (w+1)/4^s - (w+1)/5^s + w/7^s + 1/8^s + ...)*(1 - 1/2^s + 1/4^s - 1/5^s + 1/7^s - 1/8^s + ...)*(1 - (w+1)/2^s + w/4^s + w/5^s - (w+1)/7^s + 1/8^s + ...)*(1 - w/2^s - (w+1)/4^s + (w+1)/5^s + w/7^s - 1/8^s + ...) = 1 + 6/19^s + 6/37^s + 1/64^s + 6/73^s + ...;
Column k = 10: (1 + 1/3^s + 1/7^s + 1/9^s + ...)*(1 + i/3^s - i/7^s - 1/9^s + ...)*(1 - 1/3^s - 1/7^s + 1/9^s + ...)*(1 - i/3^s + i/7^s - 1/9^s + ...) = 1 + 4/11^s + 4/31^s + 4/41^s + 4/61^s + 4/71^s + 1/81^s + 4/101^s + ...

Jianing Song, Table of n, a(n) for n = 0..11324 (the first 150 diagonals, with n+k = 1..150)

Columns: A000012 (k=1 and k=2), A033687 (k=3), A008441 (k=4), A378008 (k=5), A097195 (k=6), A378009 (k=7), A378010 (k=8), A378011 (k=9), A378012 (k=10).

Cf. A378006.

A378007(n,k) = {
my(f = factor(k*n+1), res = 1); for(i=1, #f~, my(d = znorder(Mod(f[i,1],k)));
if(f[i,2] % d != 0, return(0), my(m = f[i,2]/d, r = eulerphi(k)/d); res *= binomial(m+r-1,r-1)));
res;}

See A378006.

For odd k, T(2*k,n) = T(k,2*n).

A045833 Expansion of eta(q^9)^3 / eta(q^3) in powers of q.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + 2*q^31 + ...

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A033685, A033687, A097195, A217219.

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^9]^3 / QPochhammer[ q^3], {q, 0, n}]; (* Michael Somos, Feb 22 2015 *)
f[p_, e_] := If[Mod[p, 3] == 1, e + 1, (1 + (-1)^e)/2]; f[3, e_] := 0; a[0] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Oct 13 2022 *)

{a(n) = local(A, p, e); if( n<0, 0, A=factor(n); prod(k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p!=3, if( p%3==1, e+1, !(e%2))))))}; \\ Michael Somos, May 25 2005

{a(n) = local(A); if( (n<1) || (n%3!=1), 0, n = (n-1)/3; A = x * O(x^n); polcoeff( eta(x^3 + A)^3 / eta(x + A), n))}; \\ Michael Somos, May 25 2005

From Michael Somos, May 25 2005: (Start)
Euler transform of period 9 sequence [ 0, 0, 1, 0, 0, 1, 0, 0, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*w - 2*u*w^2 - v^3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1*u3^2 + u1*u6^2 - u1*u3*u6 - u2^2*u3.
a(3*n) = a(3*n + 2) = 0. a(3*n + 1) = A033687(n). a(6*n + 1) = A097195(n). 3*a(n) = A033685(n).
Multiplicative with a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1+(-1)^e)/2 if p == 2 (mod 3).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2*u3^2 + 2*u2*u3*u6 + 4*u2*u6^2 - u1^2*u6. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(9*sqrt(3)) = 0.403066... . - Amiram Eldar, Oct 13 2022
Dirichlet g.f.: L(chi_1,s)*L(chi_{-1},s), where chi_1 = A011655 and chi_{-1} = A102283 are respectively the principal and the non-principal Dirichlet character modulo 3. For the formula of the sequence whose Dirichlet g.f. is Product_{chi} L(chi,s), where chi runs through all Dirichlet characters modulo k, see A378006. This sequence is the case k = 3. - Jianing Song, Nov 13 2024

A097190 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/3) = 9^n, where R_n(y) forms the initial (n+1) terms of g.f. A097191(y)^(n+1).

Original entry on oeis.org

1, 1, 24, 1, 36, 612, 1, 48, 1104, 15912, 1, 60, 1740, 32130, 417690, 1, 72, 2520, 56700, 912492, 11027016, 1, 84, 3444, 91350, 1750014, 25562628, 292215924, 1, 96, 4512, 137808, 3059856, 52303968, 710025264, 7764594552, 1, 108, 5724, 197802, 4992354

Offset: 0

Paul D. Hanna, Aug 03 2004

			Row polynomials evaluated at y=1/3 equals powers of 9:
9^1 = 1 + 24/3;
9^2 = 1 + 36/3 + 612/3^2;
9^3 = 1 + 48/3 + 1104/3^2 + 15912/3^3;
9^4 = 1 + 60/3 + 1740/3^2 + 32130/3^3 + 417690/3^4;
where A097191(y)^(n+1) has the same initial terms as the n-th row:
A097191(y) = 1 + 12y + 60y^2 + 90y^3 - 558y^4 - 2916y^5 + 2160y^6 +...
A097191(y)^2 = 1 + 24y +...
A097191(y)^3 = 1 + 36y + 612y^2 +...
A097191(y)^4 = 1 + 48y + 1104y^2 + 15912y^3 +...
A097191(y)^5 = 1 + 60y + 1740y^2 + 32130y^3 + 417690y^4 +...
Rows begin with n=0:
  1;
  1, 24;
  1, 36,  612;
  1, 48, 1104,  15912;
  1, 60, 1740,  32130,  417690;
  1, 72, 2520,  56700,  912492, 11027016;
  1, 84, 3444,  91350, 1750014, 25562628, 292215924;
  1, 96, 4512, 137808, 3059856, 52303968, 710025264, 7764594552; ...

G. C. Greubel, Rows n = 0..50 of triangle, flattened

Cf. A097186, A097191, A097192, A097193, A097194, A097195.

Table[SeriesCoefficient[3*y/((1-27*x*y) + (3*y-1)*(1-27*x*y)^(8/9)), {x, 0,n}, {y,0,k}], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 17 2019 *)

{T(n,k)=if(n==0,1,if(k==0,1,if(k==n, 3^n*(9^n-sum(j=0,n-1, T(n,j)/3^j)), polcoeff((Ser(vector(n,i,T(n-1,i-1)),x) +x*O(x^k))^((n+1)/n),k,x))))}

G.f.: A(x, y) = 3*y/((1-27*x*y) + (3*y-1)*(1-27*x*y)^(8/9)).

G.f.: A(x, y) = A097192(x*y)/(1 - x*A097193(x*y)).

A374900 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+1)).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jul 31 2024

nonn

Cf. A008441, A033687, A097195, A340456.

my(N=110, x='x+O('x^N)); Vec(sum(k=-N, N, x^k/(1-x^(7*k+1))))

my(N=110, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^2*(1-x^(7*k-2))*(1-x^(7*k-5))/((1-x^(7*k-1))*(1-x^(7*k-6)))^2))

G.f.: Product_{k>0} (1-x^(7*k))^2 * (1-x^(7*k-2)) * (1-x^(7*k-5)) / ((1-x^(7*k-1)) * (1-x^(7*k-6)))^2.

A115978 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, 6, 0, 0, -4, 0, -2, 0, 0, 6, -4, 0, 0, 6, 0, 0, -4, 0, -4, 0, 0, 0, -2, 0, -2, 12, 0, 0, -4, 0, 0, 0, 0, 6, -4, 0, -4, 0, 0, 0, -4, 0, 0, 0, 0, 6, -6, 0, 0, 12, 0, 0, 0, 0, -4, 0, 0, 0, -4, 0, -4, 6, 0, 0, -4, 0, 0, 0, 0, 0, -4, 0, -2, 12, 0, 0, -4, 0, -2, 0, 0, 12, 0, 0, 0, 0, 0, 0, -8, 0, -4, 0, 0, 0, -4, 0, 0, 6, 0

Offset: 0

Michael Somos, Feb 09 2006

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 - 2*q - 2*q^3 + 6*q^4 - 4*q^7 - 2*q^9 + 6*q^12 - 4*q^13 + ...

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. A004016, A033716, A033762, A097195, A112604, A122861.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3], {q, 0, n}] (* Michael Somos, Nov 09 2013 *)

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

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); -2 * prod( k=1, matsize(A)[1], if(p = A[k,1], e = A[k,2]; if( p==2, -3 * ((e+1)%2), if( p==3, 1, if( p%6==1, e+1, (e+1)%2))))))} /* Michael Somos, Nov 09 2013 */

Expansion of theta_4(q) * theta_4(q^3) in powers of q.
Expansion of (4 * a(q^4) - a(q)) / 3 = (4 * b(q^4) - b(q)) * b(q) / (3 * b(q^2)) in powers of q where a(), b() are cubic AGM theta functions. - Michael Somos, Nov 09 2013
Expansion of (eta(q) * eta(q^3))^2 / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ -2, -1, -4, -1, -2, -2, ...].
Moebius transform is period 12 sequence [ -2, 2, 0, 6, 2, 0, -2, -6, 0, -2, 2, 0, ...]. - Michael Somos, Nov 09 2013
a(n) = -2*b(n) where b(n) is multiplicative and b(2^e) = -3 * (1 + (-1)^e) / 2 if e>0, b(3^e) = 1, b(p^e) = 1+e if p == 1 (mod 6), b(p^e) = (1 +(-1)^e) / 2 if p == 5 (mod 6).
Given g.f. A(x), then B(x) = A(x)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v*(u + v)^2 - 4*u * (w^2 - v*w + v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 192^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033762. - Michael Somos, Nov 09 2013
G.f.: 1 - 2*(Sum_{k>0} x^k / (1 + x^k + x^(2*k)) - 4 * x^(4*k) / (1 + x^(4*k) + x^(8*k))).
G.f.: (Sum_{k in Z} (-x)^(k^2)) * (Sum_{k in Z} (-x)^(3*k^2)).
a(n) = -2 * A115979(n) unless n=0. a(n) = (-1)^n * A033716(n).
a(3*n + 2) = a(4*n + 2) = 0. a(3*n) = a(n). a(2*n + 1) = -2 * A033762(n). a(3*n + 1) = -2 * A122861(n). a(4*n) = A004016(n). a(4*n + 1) = -2 * A112604(n). a(6*n + 1) = -2 * A097195(n). - Michael Somos, Nov 09 2013

A123530 Expansion of q^(-1/2)eta(q)^2eta(q^6)^3/(eta(q^2)*eta(q^3)^2) in powers of q.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 02 2006

sign

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

QP = QPochhammer; s = QP[q]^2*(QP[q^6]^3/(QP[q^2]*QP[q^3]^2)) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

{a(n)=if(n<0, 0, n=2*n+1; sumdiv(n, d, kronecker(-12,d)*[0,1,0,-2,0,1][n/d%6+1]))}

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

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

Euler transform of period 6 sequence [ -2, -1, 0, -1, -2, -2, ...].
a(n) = b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = -2 if e>0, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} F(x^(6k-5))-F(x^(6k-1)) where F(x)=(x-x^3)/(1+x^2+x^4).
a(3*n+2) = 0.
a(3*n) = A097195(n).
a(3*n+1) = -2*A033762(n).
a(n) = A097109(2*n+1) = A112848(2*n+1).

A153728 Expansion of q^(-1/3) * (eta(q)^8 + 8 * eta(q^4)^8) in powers of q^2.

Original entry on oeis.org

1, 20, -70, 56, -125, 308, 110, -520, 57, 0, 182, -880, 1190, 884, 0, -1400, -1330, 1820, -646, 0, -1331, 380, 1120, 2576, 0, 1748, -3850, -3400, 2703, -2500, 3458, 0, -1150, -5236, 0, 6032, 6160, -3220, 4466, 0, -7378, -3920, 0, 2200, 0, 812, -4030, 5600, -4913

Offset: 0

Michael Somos, Dec 31 2008

sign

This is a member of an infinite family of integer weight modular forms. g_1 = A097195, g_2 = A000727, g_3 = A152243, g_4 = A153728. - Michael Somos, Jun 10 2015

			G.f. = 1 + 20*x - 70*x^2 + 56*x^3 - 125*x^4 + 308*x^5 + 110*x^6 - 520*x^7 + ...
G.f. = q + 20*q^7 - 70*q^13 + 56*q^19 - 125*q^25 + 308*q^31 + 110*q^37 + ...

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

Cf. A000727, A000731, A097195, A152243, A153729, A161969.

A := Basis( CuspForms( Gamma0(36), 4), 289); A[1] + 20*A[7] - 70*A[12]; /* Michael Somos, Jun 10 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^8 + 8 x QPochhammer[ x^4]^8, {x, 0, 2 n}]; (* Michael Somos, Jun 10 2015 *)

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

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 6*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%6==5, if( e%2, 0, (-p)^(3*e/2)), for(x=1, sqrtint(p\3), if( issquare(p-3*x^2, &y), break)); if( y%3!=1, y=-y); y*=2; y = y^3 - 3*p*y; a0=1; a1=y; for(i=2, e, x = y*a1 - p^3*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Jun 10 2015 */

a(n) = b(6*n + 1) where b(n) is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * (-1)^(e/2) * p^(3*e/2) if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - b(p^(e-2)) * p^3 if p == 1 (mod 6) where b(p) = (x^2 - 3*p)*x, 4*p = x^2 + 3*y^2, |x| < |y| and x == 2 (mod 3).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 648 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A153729.
a(n) = A000731(2*n) = A153729(2*n) = A161969(2*n). - Michael Somos, Jun 10 2015

A112298 Expansion of (a(q) - 3a(q^2) + 2a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, -3, 1, 3, 0, -3, 2, -3, 1, 0, 0, 3, 2, -6, 0, 3, 0, -3, 2, 0, 2, 0, 0, -3, 1, -6, 1, 6, 0, 0, 2, -3, 0, 0, 0, 3, 2, -6, 2, 0, 0, -6, 2, 0, 0, 0, 0, 3, 3, -3, 0, 6, 0, -3, 0, -6, 2, 0, 0, 0, 2, -6, 2, 3, 0, 0, 2, 0, 0, 0, 0, -3, 2, -6, 1, 6, 0, -6, 2, 0, 1, 0, 0, 6, 0, -6, 0, 0, 0, 0, 4, 0, 2, 0, 0, -3, 2, -9, 0, 3, 0, 0, 2, -6, 0

Offset: 1

Michael Somos, Sep 02 2005

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. = q - 3*q^2 + q^3 + 3*q^4 - 3*q^6 + 2*q^7 - 3*q^8 + q^9 + 3*q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A093602, A093829, A097195, A112604, A112605, A129576, A244375.

Cf. A000122, A000700, A004016, A010054, A005882, A005928, A121373.

A := Basis( ModularForms( Gamma1(12), 1), 106); A[2] - 3*A[3] + A[4] + 3*A[5]; /* Michael Somos, Jan 17 2015 */

a[ n_] := SeriesCoefficient[ q QPochhammer[ q, q^2]^3 QPochhammer[ -q^6, q^6]^3 EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(3/2)] / (2 q^(3/8)), {q, 0, n}]; (* Michael Somos, Jan 17 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, JacobiSymbol[ -3, n/#] {1, -2, 1, 0}[[Mod[#, 4, 1]]] &]]; (* Michael Somos, Jan 17 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-3, n/d)*[0, 1, -2, 1][d%4 + 1]))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^12 + A))^3/ (eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^6 + A)), n))};

From Michael Somos, Jan 17 2015: (Start)
Expansion of b(q) * (b(q^4) - b(q)) / (3*b(q^2)) in powers of q  where b() is a cubic AGM theta function.
Expansion of q * chi(-q)^3 * phi(-q^2) * psi(q^3) / chi(-q^6)^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q * phi(-q)^2 * psi(q^6)^2 / (psi(-q) * psi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of q * f(q) * f(-q, -q^5)^4 / f(q^3)^3 in powers of q where f() is a Ramanujan theta function. (End)
Expansion of (eta(q) * eta(q^12))^3 / (eta(q^2) * eta(q^3) * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ -3, -2, -2, -1, -3, 0, -3, -1, -2, -2, -3, -2, ...].
Moebius transform is period 12 sequence [ 1, -4, 0, 6, -1, 0, 1, -6, 0, 4, -1, 0, ...].
Multiplicative with a(2^e) = 3(-1)^e if e>0, a(3^e)=1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 2 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker(-3, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)).
a(n) = -(-1)^n * A244375(n). a(6*n + 5) = 0, a(3*n) = a(n).
a(2*n) = -3 * A093829(n). a(2*n + 1) = A033762(n). a(3*n + 1) = A129576(n). a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(6*n + 2) = -3 * A033687(n).
Sum_{k=1..n} abs(a(k)) ~ (Pi/sqrt(3)) * n. - Amiram Eldar, Jan 23 2024

A112848 Expansion of eta(q)eta(q^2)eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 22 2005

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A033687, A033762, A092829, A093829, A097195, A248897, A255648 (absolute values).

QP = QPochhammer; s = QP[q]*QP[q^2]*(QP[q^18]^2/(QP[q^6]*QP[q^9])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)
f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[2, e_] := (-1)^e; f[3, e_]:= -2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 28 2024 *)

{a(n)=if(n<1, 0, if(n%3==0, n/=3; -2,1)* sumdiv(n,d,kronecker(-12,d) -if(d%2==0, 2*kronecker(-3,d/2))))}

{a(n)=local(A); if (n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^2+A)*eta(x^18+A)^2/ eta(x^6+A)/eta(x^9+A), n))}

Euler transform of period 18 sequence [ -1, -2, -1, -2, -1, -1, -1, -2, 0, -2, -1, -1, -1, -2, -1, -2, -1, -2, ...].
Moebius transform is period 18 sequence [1, -2, -3, 2, -1, 6, 1, -2, 0, 2, -1, -6, 1, -2, 3, 2, -1, 0, ...].
Multiplicative with a(2^e) = (-1)^e, 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).
G.f.: Sum_{k>0} Kronecker(-3, k) x^k(1-x^(2k))^2/(1-x^(6k)) = x Product_{k>0} (1-x^k)(1-x^(2k))(1+x^(9k))(1+x^(6k)+x^(12k)).
a(3n) = -2*A092829(n). a(3n+1) = A093829(3n+1) = A033687(n). a(3n+2) = A093829(3n+2). a(6n)/2 = A093829(n). a(6n+1) = A097195(n). a(6n+3) = -2*A033762(n). a(6n+5) = 0.
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 23 2024

cp's OEIS Frontend

A227696 Expansion of f(x^3)^3 / f(x) in powers of x where f() is a Ramanujan theta function.

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A378007 Square table read by descending antidiagonals: T(n,k) = A378006(k*n+1,k).

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A045833 Expansion of eta(q^9)^3 / eta(q^3) in powers of q.

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A097190 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/3) = 9^n, where R_n(y) forms the initial (n+1) terms of g.f. A097191(y)^(n+1).

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A374900 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+1)).

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A115978 Expansion of phi(-q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

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A123530 Expansion of q^(-1/2)*eta(q)^2*eta(q^6)^3/(eta(q^2)*eta(q^3)^2) in powers of q.

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A123530 Expansion of q^(-1/2)eta(q)^2eta(q^6)^3/(eta(q^2)*eta(q^3)^2) in powers of q.

A112298 Expansion of (a(q) - 3a(q^2) + 2a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.

A112848 Expansion of eta(q)eta(q^2)eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.