A030203 -id:A030203 - cp's OEIS frontend

A209940 Expansion of psi(x^4) * phi(-x^4)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta function.

1, -2, 4, -8, 7, -10, 12, -8, 18, -18, 16, -24, 21, -20, 28, -32, 20, -32, 36, -24, 42, -42, 28, -48, 57, -36, 52, -40, 36, -58, 60, -56, 48, -66, 48, -72, 74, -42, 80, -80, 61, -82, 72, -56, 90, -96, 64, -72, 98, -70, 100, -104, 64, -106, 108, -72, 114, -96

Offset: 0

Michael Somos, Mar 16 2012

sign

Number 47 of the 74 eta-quotients listed in Table I of Martin (1996).

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

			G.f. = 1 - 2*x + 4*x^2 - 8*x^3 + 7*x^4 - 10*x^5 + 12*x^6 - 8*x^7 + 18*x^8 + ...
G.f. = q - 2*q^3 + 4*q^5 - 8*q^7 + 7*q^9 - 10*q^11 + 12*q^13 - 8*q^15 + 18*q^17 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 44.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A113417, A113419, A258096.

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^4]^9 / (QPochhammer[ q^2]^5 QPochhammer[ q^8]^2), {q, 0, n}]; (* Michael Somos, May 19 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^9 / (eta(x^2 + A)^5 * eta(x^8 + A)^2), n))};

{a(n) = my(A, p, e, f); if( n<0, 0, A = factor(2*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0^e, p==3, ((-p)^(e+1) - 1) / ((-p) - 1), p *= kronecker( 18, p); (-1)^(e*(p\6)) * (p^(e+1) - 1) / (p - 1))))};

Expansion of q^(-1/2) * eta(q)^2 * eta(q^4)^9 / (eta(q^2)^5 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, 3, -2, -6, -2, 3, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 512^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113419.
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = (1 - (-3)^(e+1)) / 4, b(p^e) = (-1)^(e * [p/6]) * ((p*f)^(e+1) - 1) / (p*f - 1) where f = Kronecker( 18, p).
a(n) = (-1)^n * A258096(n) = (-1)^floor(n/2) * A113419(n) = (-1)^(n + floor(n/2)) * A113417(n).

A209942 Expansion of (psi(-x) * phi(x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 14, 81, 238, 322, 0, -429, -82, 0, -2162, -3038, 1134, 2401, -2482, 0, 6958, 3332, 0, 1442, 0, 6561, 4508, -9758, 0, -1918, -18802, 0, -9362, -24638, 19278, 14641, -14756, 0, 0, 6562, 0, -1148, 33998, 26082, 20398, 0, 0, 28083, -49042, 0, 64078, -30268, 0

Offset: 0

Michael Somos, Mar 16 2012

sign

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

Number 60 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 + 14*x + 81*x^2 + 238*x^3 + 322*x^4 - 429*x^6 - 82*x^7 - 2162*x^9 + ...
G.f. = q + 14*q^5 + 81*q^9 + 238*q^13 + 322*q^17 - 429*q^25 - 82*q^29 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000143, A134343, A258771.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^19 / (QPochhammer[ x] QPochhammer[ x^4])^7)^2, {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^19 / (eta(x + A) * eta(x^4 + A) )^7 )^2, n))};

Expansion of q^(-1/4) * ( eta(q^2)^19 / (eta(q) * eta(q^4) )^7 )^2 in powers of q.
Euler transform of period 4 sequence [ 14, -24, 14, -10, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 32768 (t/i)^5 f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) otherwise.
a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = 81 * a(n). Convolution of A000143 and A134343.
Convolution square of A258771. - Michael Somos, Jun 09 2015

A030202 Expansion of q^(-1/4) * eta(q) * eta(q^5) in powers of q.

Original entry on oeis.org

1, -1, -1, 0, 0, 0, 1, 2, 0, 0, -2, 1, -1, 0, 0, -2, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, -2, 0, 0, 1, -1, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, -1, 2, 0, 0, -2, 1, 0, 0, 0, -2, 0, -2, 0, 0, -2, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 0, 0, -2, 0, 0, 2, -1, -2, 0, 0

Offset: 0

N. J. A. Sloane

Number 62 of the 74 eta-quotients listed in Table I of Martin (1996).

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

			G.f. = 1 - x - x^2 + x^6 + 2*x^7 - 2*x^10 + x^11 - x^12 - 2*x^15 + x^20 + ...
G.f. = q - q^5 - q^9 + q^25 + 2*q^29 - 2*q^41 + q^45 - q^49 - 2*q^61 + q^81 + ...

Bruce Berndt, Ramanujan's Notebooks Part III, Springer-Verlag; see page 44.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A030205, A030210, A159818.

Basis( CuspForms( Gamma1(80), 1), 413)[1]; /* Michael Somos, May 16 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^5], {x, 0, n}] (* Michael Somos, Aug 08 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 1, Pi/5, q^2] EllipticTheta[ 1, 2 Pi/5, q^2] / Sqrt[5], {q, 0, 4 n + 1}] // FullSimplify; (* Michael Somos, Aug 08 2011 *)

{a(n) = if( n<0, 0, polcoeff( eta(x^5 + x * O(x^n)) * eta(x + x * O(x^n)), n))}; /* Michael Somos, Sep 04 2007 */

{a(n) = my(A, p, e, x, y); if( n<0, 0, n = 4*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p==5, (-1)^e, p%20>10, !(e%2), p%4==3, kronecker( -4, e+1), for( y=1, sqrtint(p\5), if( issquare(p - 5*y^2), x=y; break)); (-1)^(e*x) * (e+1))))}; /* Michael Somos, Sep 04 2007 */

Expansion of f(-x, -x^4) * f(-x^2, -x^3) in powers of x where f() is the Ramanujan two-variable theta function.
Expansion of q^(-1) * (phi(q) * phi(q^20) - phi(q^4) * phi(q^5)) / 2 in powers of q^4 where phi() is a Ramanujan theta function.
Euler transform of period 5 sequence [ -1, -1, -1, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 80^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(5^e) = (-1)^e, b(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20), b(p^e) = (i^n +(-i)^n)/2 if p == 3, 7 (mod 20), b(p^e) = (-1)^(e*y) * (e+1) if p == 1, 9 (mod 20) where p = x^2 + 5*y^2. - Michael Somos, Sep 04 2007
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(5*k)).
a(5*n + 3) = a(5*n + 4) = a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = -a(n). - Michael Somos, May 16 2015
Convolution square is A030205. - Michael Somos, May 16 2015
a(n) = (-1)^n * A159818(n). - Michael Somos, May 16 2015

A106406 Expansion of (eta(q) * eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q.

Original entry on oeis.org

1, -2, -1, 3, -1, 2, 0, -4, 1, 2, 0, -3, 0, 0, 1, 5, -2, -2, 2, -3, 0, 0, -2, 4, 1, 0, -1, 0, 0, -2, 2, -6, 0, 4, 0, 3, 0, -4, 0, 4, 0, 0, 0, 0, -1, 4, -2, -5, 1, -2, 2, 0, -2, 2, 0, 0, -2, 0, 0, 3, 2, -4, 0, 7, 0, 0, 0, -6, 2, 0, 0, -4, 0, 0, -1, 6, 0, 0, 2

Offset: 1

Michael Somos, May 02 2005

Number 30 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - 2*q^2 - q^3 + 3*q^4 - q^5 + 2*q^6 - 4*q^8 + q^9 + 2*q^10 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A028955, A028957, A035175, A123864.

A := Basis( ModularForms( Gamma1(15), 1), 80); A[2] - 2*A[3] - A[4] + 3*A[5] - A[6] + 2*A[7]; /* Michael Somos, May 18 2015 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^15])^2 / (QPochhammer[ q^3] QPochhammer[ q^5]), {q, 0, n}]; (* Michael Somos, May 18 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ #, 3] KroneckerSymbol[ n/#, 5] &]]; (* Michael Somos, May 18 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^15 + A)^2 / (eta(x^3 + A) * eta(x^5 + A)), n))};

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( d, 3) * kronecker( n/d, 5)))};

{a(n) = my(A, p, e, x); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==3 || p==5, (-1)^e, (p%15) != 2^(x = valuation( p%15, 2)), (e+1)%2, (e+1) * (-1)^(x*e))))};

{a(n) = if( n<1, 0, (qfrep([2, 1;1, 8],n, 1) - qfrep([4, 1;1, 4], n, 1))[n])}; /* Michael Somos, Aug 25 2006 */

Euler transform of period 15 sequence [-2, -2, -1, -2, -1, -1, -2, -2, -1, -1, -2, -1, -2, -2, -2, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 4 * u*v*w + 2 * u*w^2 + u^2*w.
a(n) is multiplicative with a(3^e) = a(5^e) = (-1)^e, a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 4 (mod 15), a(p^e) = (e+1) * (-1)^e if p == 2, 8 (mod 15). - Michael Somos, Oct 19 2005
G.f.: (1/2) * (Sum_{n,m in Z} x^(n^2 + n*m + 4*m^2) - x^(2*n^2 + n*m + 2 *m^2)). - Michael Somos, Aug 25 2006
G.f.: Sum_{k>0} Kronecker(k, 3) * x^k * (1 - x^k) * (1 - x^(2*k)) / (1 - x^(5*k)) = Sum_{k>0} Kronecker(k, 5) * x^k * (1 - x^k) / (1 - x^(3*k)).
G.f.: x * Product_{k>0} ((1 - x^k) * (1 - x^(15*k)))^2 / ((1 - x^(3*k)) * (1 - x^(5*k))).
a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0. a(3*n) = a(5*n) = -a(n).
A035175(n) = |a(n)|. a(n)>0 iff n in A028957. a(n)<0 iff n in A028955.
G.f. is a period 1 Fourier series which satisfies f(-1 / (15 t)) = 15^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, May 18 2015

A109039 Expansion of eta(q) * eta(q^3) * (eta(q^4) * eta(q^6) / eta(q^12))^2 in powers of q.

Original entry on oeis.org

1, -1, -1, -1, -1, 4, -1, 6, -1, -1, 4, -12, -1, -14, 6, 4, -1, 16, -1, 18, 4, 6, -12, -24, -1, -21, -14, -1, 6, 28, 4, 30, -1, -12, 16, -24, -1, -38, 18, -14, 4, 40, 6, 42, -12, 4, -24, -48, -1, -43, -21, 16, -14, 52, -1, 48, 6, 18, 28, -60, 4, -62, 30, 6

Offset: 0

Michael Somos, Jun 17 2005

sign

Number 25 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - q - q^2 - q^3 - q^4 + 4*q^5 - q^6 + 6*q^7 - q^8 - q^9 + 4*q^10 + ...

David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.

Cf. A109040, A124815.

A := Basis( ModularForms( Gamma1(12), 2), 64); A[1] - A[2] - A[3] - A[4] - A[5] + 4*A[6] - A[7] + 6*A[8] - A[9]; /* Michael Somos, May 18 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^3] (QPochhammer[ q^4] QPochhammer[ q^6] / QPochhammer[ q^12])^2, {q, 0, n}]; (* Michael Somos, May 18 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] QPochhammer[ q^3, q^6]^3 EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(1/2)]^2 / (4 q^(3/8)), {q, 0, n}]; (* Michael Somos, May 18 2015 *)

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

Euler transform of period 12 sequence [ -1, -1, -2, -3, -1, -4, -1, -3, -2, -1, -1, -4, ...].
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 - x^(4*k))^2 / (1 + x^(6*k))^2.
a(n) = -A109040(n) unless n=0. a(2*n) = a(3*n) = a(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(3/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A124815. - Michael Somos, May 18 2015
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(3)) = 0.237425... . - Amiram Eldar, Jan 29 2024

A111949 Expansion of eta(q) * eta(q^2) * eta(q^10) * eta(q^20) / (eta(q^4) * eta(q^5)) in powers of q.

Original entry on oeis.org

1, -1, -2, 1, 1, 2, -2, -1, 3, -1, 0, -2, 0, 2, -2, 1, 0, -3, 0, 1, 4, 0, -2, 2, 1, 0, -4, -2, 2, 2, 0, -1, 0, 0, -2, 3, 0, 0, 0, -1, 2, -4, -2, 0, 3, 2, -2, -2, 3, -1, 0, 0, 0, 4, 0, 2, 0, -2, 0, -2, 2, 0, -6, 1, 0, 0, -2, 0, 4, 2, 0, -3, 0, 0, -2, 0, 0, 0, 0, 1, 5, -2, -2, 4, 0, 2, -4, 0, 2, -3, 0, -2, 0, 2, 0, 2, 0, -3, 0, 1, 2, 0, -2, 0, 4

Offset: 1

Michael Somos, Aug 22 2005

Number 37 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - q^2 - 2*q^3 + q^4 + q^5 + 2*q^6 - 2*q^7 - q^8 + 3*q^9 - q^10 + ...

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Cf. A033764, A035170, A129391.

a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^10] QPochhammer[ q^20] / (QPochhammer[ q^4] QPochhammer[ q^5]), {q, 0, n}]; (* Michael Somos, May 19 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ Mod[d, 2] (-1)^Quotient[d, 2] KroneckerSymbol[ n/d, 5], { d, Divisors[ n]}]]; (* Michael Somos, May 19 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^10 + A) * eta(x^20 + A) / eta(x^4 + A) / eta(x^5 + A), n))};

{a(n) = if( n<1, 0, sumdiv( n, d, (d%2) * (-1)^(d\2) * kronecker( n/d, 5)))};

{a(n) = if( n<1, 0, qfrep( [1, 0; 0, 5], n)[n] - qfrep( [2, 1; 1, 3], n)[n])};

Euler transform of period 20 sequence [-1, -2, -1, -1, 0, -2, -1, -1, -1, -2, -1, -1, -1, -2, 0, -1, -1, -2, -1, -2, ...].
a(n) is multiplicative with a(p^e) = (-1)^e if p = 2, a(p^e) = 1 if p = 5, a(p^e) = (1 + (-1)^e) / 2 if p == 11, 13, 17, 19 (mod 20), a(p^e) = e + 1 if p == 1, 9 (mod 20), a(p^e) = (e + 1)*(-1)^e if p == 3, 7 (mod 20).
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 - x^k) * (1 - x^(2*k)) / (1 - x^(5*k)).
G.f.: Sum_{k>0} Kronecker(k, 5) * x^k / (1 + x^(2*k)).
G.f.: x * Product_{k>0} (1 - x^k) * (1 + x^(5*k)) * (1 - x^(20*k)) / (1 + x^(2*k)).
|a(n)| = A035170(n). a(2*n) = -a(n). a(2*n + 1) = A129391(n). a(4*n + 3) = -2 * A033764(n).
a(5*n) = a(n). - Michael Somos, May 19 2015

A113421 Expansion of eta(q)^2 * eta(q^4) * eta(q^6)^2 * eta(q^12) / eta(q^3)^2 in powers of q.

Original entry on oeis.org

1, -2, -1, 4, -4, 2, 6, -8, 1, 8, -12, -4, 14, -12, 4, 16, -16, -2, 18, -16, -6, 24, -24, 8, 21, -28, -1, 24, -28, -8, 30, -32, 12, 32, -24, 4, 38, -36, -14, 32, -40, 12, 42, -48, -4, 48, -48, -16, 43, -42, 16, 56, -52, 2, 48, -48, -18, 56, -60, 16, 62, -60, 6, 64, -56, -24, 66, -64, 24, 48, -72, -8, 74, -76, -21, 72

Offset: 1

Michael Somos, Oct 29 2005

Number 26 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - 2*q^2 - q^3 + 4*q^4 - 4*q^5 + 2*q^6 + 6*q^7 - 8*q^8 + q^9 + 8*q^10 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^2 QPochhammer[ q^4] QPochhammer[ q^6]^2 QPochhammer[ q^12] / QPochhammer[ q^3]^2, {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

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

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

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

Euler transform of period 12 sequence [-2, -2, 0, -3, -2, -2, -2, -3, 0, -2, -2, -4, ...].
G.f.: Sum_{k>0} (3*k - 2) * x^(3*k - 2) / (1 + x^(6*k - 4)) - (3*k - 1) * x^(3*k - 1) / (1 + x^(6*k - 2)).
G.f.: Sum_{k>0} -(-1)^k * x^(2*k - 1) * (1 - x^(2*k - 1))^2 * (1 - x^(4*k - 2)) / (1 - x^(6*k - 3))^2.
a(n) is multiplicative with a(2^e) = (-2)^e, a(3^e) = (-1)^e, a(p^e) = (x^(e+1) - y^(e+1)) / (x - y) where x = p * Kronecker( -3, p) and y = (-1)^[p/2].

A124340 Number of solutions to n = x^2 + 2y^2 + 4(T(z) + T(w)) + 1 where x and y are integers, z and w are nonnegative integers and T(x) = (x^2+x)/2.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 8, 8, 7, 8, 10, 8, 12, 16, 8, 16, 18, 14, 18, 16, 16, 20, 24, 16, 21, 24, 20, 32, 28, 16, 32, 32, 20, 36, 32, 28, 36, 36, 24, 32, 42, 32, 42, 40, 28, 48, 48, 32, 57, 42, 36, 48, 52, 40, 40, 64, 36, 56, 58, 32, 60, 64, 56, 64, 48, 40, 66

Offset: 1

Michael Somos, Oct 26 2006

Number 18 of the 74 eta-quotients listed in Table I of Martin (1996).

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

			G.f. = q + 2*q^2 + 2*q^3 + 4*q^4 + 4*q^5 + 4*q^6 + 8*q^7 + 8*q^8 + 7*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A091337, A328895.

Cf. A000122, A000700, A010054, A121373.

with(numtheory):
A091337 := n -> [0, 1, 0, -1, 0, -1, 0, 1][`mod`(n, 8)+1]:
seq(add(A091337(n/d)d, d in divisors(n)), n = 1..60); # Peter Bala, Jan 06 2021

a[n_] := Sum[JacobiSymbol[2, d]*n/d, {d, Divisors[n]}]; a /@ Range[80] (* Jean-François Alcover, Jan 10 2014 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^3 QPochhammer[ q^4] QPochhammer[ q^8]^2 / QPochhammer[ q]^2, {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

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

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

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^8 + A)^2 / eta(x + A)^2, n))};

Expansion of q * phi(q) * phi(q^2) * psi(q^4)^2 in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2)^3 * eta(q^4) * eta(q^8)^2 / eta(q)^2 in powers of q.
Euler transform of period 8 sequence [ 2, -1, 2, -2, 2, -1, 2, -4, ...].
a(n) is multiplicative with a(2^e) = 2^e, a(p^e) = (p^(e+1) - 1)/(p - 1) if p == 1, 7 (mod 8), a(p^e) = (p^(e+1) + (-1)^e)/(p + 1) if p == 3, 5 (mod 8).
G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
G.f.: x * Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k)) * (1 - x^(4*k)) * (1 - x^(8*k))^2.
From Peter Bala, Jan 06 2021: (Start)
a(n) = Sum_{ d | n } X(n/d)*d, where X(k) = A091337(k) is a non-principal Dirichlet charcter modulo 8.
G.f.: A(x) = Sum_{n = -oo..oo} (-1)^n*x^(4*n+1)/(1 - x^(4*n+1))^2. (End)
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A328895. - Amiram Eldar, Feb 20 2024

A125095 Expansion of phi(-x) * psi(x^4) in powers of x where psi(), phi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 0, 0, 3, -2, 0, 0, 2, -2, 0, 0, 1, -4, 0, 0, 4, 0, 0, 0, 2, -2, 0, 0, 1, -4, 0, 0, 4, -2, 0, 0, 0, -2, 0, 0, 2, -2, 0, 0, 5, -2, 0, 0, 2, 0, 0, 0, 2, -6, 0, 0, 0, -2, 0, 0, 2, 0, 0, 0, 3, -4, 0, 0, 4, -2, 0, 0, 2, -2, 0, 0, 0, -2, 0, 0, 6, 0, 0, 0, 0

Offset: 0

Michael Somos, Nov 20 2006

sign

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

Number 45 of the 74 eta-quotients listed in Table I of Martin (1996). - Michael Somos, Mar 14 2012

			G.f. = 1 - 2*x + 3*x^4 - 2*x^5 + 2*x^8 - 2*x^9 + x^12 - 4*x^13 + 4*x^16 + ...
G.f. = q - 2*q^3 + 3*q^9 - 2*q^11 + 2*q^17 - 2*q^19 + q^25 - 4*q^27 + 4*q^33 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A033761, A113411, A112603.

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, If[ Mod[#, 8] > 3, -1, 1] &]]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^8])^2 / (QPochhammer[ x^2] QPochhammer[ x^4]), {x, 0, n}]; (* Michael Somos, Jul 09 2015 *)

{a(n) = if( n<0, 0, (-1)^n * sumdiv( 2*n + 1, d, (-1)^(d%8>3)))};

{a(n) = if( n<0, 0, n = 2*n + 1; qfrep( [1, 0; 0, 8], n)[n] - qfrep( [3, 1; 1, 3], n)[n])};

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

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

Expansion of q^(-1/2) * (eta(q)^2 * eta(q^8)^2) / (eta(q^2) * eta(q^4)) in powers of q.
Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1^2*u6 * (u1 + 3*u3) + 2 * u2^2*u3 * (u2 + 3*u6) - 3 * u3^2*u2 * (u1 + u3) - 6 * u6^2*u1 * (u2 + u6).
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (e+1) * (-1)^e if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e) / 2 if p == 5, 7 (mod 8).
Euler transform of period 8 sequence [ -2, -1, -2, 0, -2, -1, -2, -2, ...].
G.f.: (Sum_{k in Z} (-1)^k * x^k^2) * (Sum_{k>=0} x^(2*k^2 + 2*k)).
a(4*n + 2) = a(4*n + 3) = 0. a(n) = (-1)^n * A113411(n). a(4*n) = A112603(n). a(4*n + 1) = -2 * A033761(n).

A128711 Expansion of phi(x) * psi(x^4) * phi(-x^4)^4 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, 0, -5, -14, 0, 0, 2, 34, 0, 0, 25, -28, 0, 0, -28, 0, 0, 0, -46, -14, 0, 0, 49, 4, 0, 0, 68, 82, 0, 0, 0, -62, 0, 0, -142, 50, 0, 0, -11, -158, 0, 0, 146, 0, 0, 0, -94, 70, 0, 0, 0, 178, 0, 0, 98, 0, 0, 0, 75, -92, 0, 0, -28, -62, 0, 0, -238, -206, 0

Offset: 0

Michael Somos, Mar 24 2007

sign

Number 48 of the 74 eta-quotients listed in Table I of Martin (1996).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). - Michael Somos, Mar 14 2012

			G.f. = 1 + 2*x - 5*x^4 - 14*x^5 + 2*x^8 + 34*x^9 + 25*x^12 - 28*x^13 + ...
G.f. = q + 2*q^3 - 5*q^9 - 14*q^11 + 2*q^17 + 34*q^19 + 25*q^25 - 28*q^27 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A128712, A128713.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^4])^5 / (QPochhammer[ x] QPochhammer[ x^8])^2, {x, 0, n}]; (* Michael Somos, Jul 09 2015 *)

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%8>4, if( e%2, 0, p^e), for( i=1, sqrtint(p\2), if( issquare(p - 2*i^2, &x), break)); a0=1; a1=y=2*(2*x^2 - p) * (-1)^((p-1)/2); for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^4 + A))^5 / (eta(x + A) * eta(x^8 + A))^2, n))};

Expansion of q^(-1/2) * (eta(q^2) * eta(q^4))^5 / (eta(q) * eta(q^8))^2 in powers of q. - Michael Somos, Mar 14 2012
Euler transform of period 8 sequence [ 2, -3, 2, -8, 2, -3, 2, -6, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * p^e if p == 5, 7 (mod 8), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) if p == 1, 3 (mod 8) where b(p) = 2*(2*x^2 - p) * (-1)^((p-1)/2) and p = x^2 + 2*y^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(15/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^k)^6 * (1 + x^k)^8 * (1 + x^(2*k))^3 / (1 + x^(4*k))^2.
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A128712(n). a(4*n + 1) = 2 * A128713(n).

cp's OEIS Frontend

A209940 Expansion of psi(x^4) * phi(-x^4)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta function.

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A209942 Expansion of (psi(-x) * phi(x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

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A030202 Expansion of q^(-1/4) * eta(q) * eta(q^5) in powers of q.

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A106406 Expansion of (eta(q) * eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q.

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A109039 Expansion of eta(q) * eta(q^3) * (eta(q^4) * eta(q^6) / eta(q^12))^2 in powers of q.

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A111949 Expansion of eta(q) * eta(q^2) * eta(q^10) * eta(q^20) / (eta(q^4) * eta(q^5)) in powers of q.

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A124340 Number of solutions to n = x^2 + 2y^2 + 4(T(z) + T(w)) + 1 where x and y are integers, z and w are nonnegative integers and T(x) = (x^2+x)/2.