A008441 -id:A008441 - cp's OEIS frontend

A134343 Expansion of psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.

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

Offset: 0

Michael Somos, Oct 21 2007

sign

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

Number 57 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 2*x + x^2 - 2*x^3 + 2*x^4 + 3*x^6 - 2*x^7 - 2*x^9 + 2*x^10 + ...
G.f. = q - 2*q^5 + q^9 - 2*q^13 + 2*q^17 + 3*q^25 - 2*q^29 - 2*q^37 + ...

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. A008441, A053692, A113407, A121613, A143379, A209942, A213791, A246862, A246683, A246950, A259285, A259287.

A := Basis( ModularForms( Gamma1(64), 1), 321); A[2] - 2*A[6] + A[10] - 2*A[14] + 2*A[18] + 3*A[26] - 2*A[30] + 2*A[35] - 2*A[36]; /* Michael Somos, Jun 22 2015 */;

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 4 n + 1, (-1)^Quotient[#, 2] &]]; (* Michael Somos, Jun 22 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(1/2)]^2 / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, Jun 22 2015 *)
a[ n_] := SeriesCoefficient[(QPochhammer[ x] QPochhammer[ x^4] / QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Jun 22 2015 *)

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

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

Expansion of q^(-1/4) * (eta(q) * eta(q^4) / eta(q^2))^2 in powers of q.
Euler transform of period 4 sequence [ -2, 0, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 8 (t/i) f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 8), b(p^e) = (-1)^e * (e+1) if p == 5 (mod 8).
G.f.: (Product_{k>0} (1 - x^k) * (1 + x^(2*k)))^2.
a(9*n + 5) = a(9*n + 8) = 0. a(n) = (-1)^n * A008441(n). a(2*n) = A113407(n). a(2*n + 1) = -2 * A053692(n).
2 * a(n) = A204531(4*n + 1) = - A246950(n). a(4*n) = A246862(n). a(4*n + 2) = A246683(n). - Michael Somos, Jun 22 2015
a(4*n + 1) = -2 * A259287(n). a(4*n + 3) = -2 * A259285(n). - Michael Somos, Jun 24 2015
Convolution square is A121613. Convolution cube is A213791. Convolution with A000009 is A143379. Convolution with A000143 is A209942. Michael Somos, Jun 22 2015
G.f.: Sum_{k>0 odd} (x^k + x^(3*k)) / (1 + x^(4*k)) * (-1)^floor((k+1) / 4). - Michael Somos, Jun 22 2015
G.f.: Sum_{k>0 odd} (x^k - x^(3*k)) / (1 + x^(4*k)) * (-1)^floor(k / 4). - Michael Somos, Jun 22 2015

A008442 Expansion of Jacobi theta constant (theta_2(2z))^2/4.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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

a(n) is the number of ways of writing 2n as the sum of two odd positive squares. (Cf. A290081 & A008441). - Antti Karttunen, Jul 24 2017

			G.f. = q + 2*q^5 + q^9 + 2*q^13 + 2*q^17 + 3*q^25 + 2*q^29 + 2*q^37 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.26).

T. D. Noe, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A008441, A019675.

Even bisection of A290081.

A := Basis( ModularForms( Gamma1(16), 1), 106); A[2] + 2*A[6]; /* Michael Somos, Feb 22 2015 */

a[n_] := Sum[{0, 1, -1, -1, 0, 1, 1, -1}[[Mod[d, 8] + 1]], {d, Divisors[n]}]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, May 15 2013, after Michael Somos *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^2]^2 / 4, {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)
a[ n_] := If[ n < 1 || Mod[n, 4] != 1, 0, Sum[ KroneckerSymbol[ 4, d], {d, Divisors @n}]]; (* Michael Somos, Feb 22 2015 *)

{a(n) = if( n<1 || n%4!=1, 0, sumdiv(n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Apr 24 2004 */

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, -1, -1, 0, 1, 1, -1][d%8+1]))}; /* Michael Somos, Apr 24 2004 */

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^8 + A)^4 / eta(x^4 + A)^2, n))}; /* Michael Somos, Apr 24 2004 */

from sympy import divisors
def A008442(n): return 0 if n&3!=1 else sum(((a:=d&3)==1)-(a==3) for d in divisors(n,generator=True)) # Chai Wah Wu, May 17 2023

Fine gives an explicit formula for a(n) in terms of the divisors of n.
a(n) = number of divisors of n of form 8n+1, 8n+5, 8n+6 minus number of divisors of form 8n+2, 8n+3, 8n+7. [I think Fine's version is simpler - N. J. A. Sloane]
G.f.: s(8)^4/(s(4)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
Expansion of q * psi(q^4)^2 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Feb 22 2015
Expansion of eta(q^8)^4 / eta(q^4)^2 in powers of q.
Euler transform of period 8 sequence [ 0, 0, 0, 2, 0, 0, 0, -2, ...]. - Michael Somos, Apr 24 2004
a(n)=0 unless n=4k+1 in which case a(n) is the difference between number of divisors of n of form 4k+1 and 4k+3.
Multiplicative with a(2^e) = 0^e, a(p^e) = (1 + (-1)^e)/2 if p==3 mod 4 otherwise a(p^e) = 1+e. - Michael Somos, Sep 18 2004
Moebius transform is period 8 sequence [ 1, -1, -1, 0, 1, 1, -1, 0, ...]. - Michael Somos, Sep 02 2005
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k / (1 - x^(2*k)) = Sum_{k>0} x^(2*k - 1) / (1 + x^(4*k - 2)). - Michael Somos, Sep 20 2005
G.f.: Sum_{k>0} x^k * (1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) / (1 - x^(8*k)) = x Product_{k>0} (1 - x^(8*k))^4 / (1 - x^(4*k))^2. - Michael Somos, Apr 24 2004
a(4*n + 1) = A008441(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/8 = 0.392699... (A019675). - Amiram Eldar, Oct 23 2022
Dirichlet g.f.: L(chi_1,s)*L(chi_{-1},s) = L(chi_s)*beta(s), where chi_1 = A000035 and chi_{-1} = A101455 are respectively the principal and the non-principal Dirichlet character modulo 4, and beta(s) is the Dirichlet beta function. 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 = 4. - Jianing Song, Nov 13 2024

A125061 Expansion of psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Nov 18 2006

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

			G.f. = 1 + q + q^2 + 3*q^3 + q^4 + 2*q^5 + 3*q^6 + q^8 + q^9 + 2*q^10 + 3*q^12 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.53).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002175, A008441, A019669, A053692, A113407, A121444, A122856, A122857, A122865, A125061, A138741, A138745, A138746, A138949, A138950, A138951, A138952, A163746.

Cf. A000122, A000700, A010054, A121373.

s = (EllipticTheta[3, 0, q]^2 + 3*EllipticTheta[3, 0, q^3]^2)/4 + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 07 2015, from 2nd formula *)

{a(n) = if( n<1, n==0, sumdiv(n, d, ((d%2) * ((d%3==0)+1)) * (-1)^(d\6)))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1],
     [p, e] = A[k, ]; if( p==2, 1, p==3, 1+e%2*2, p%4==1, e+1, !(e%2) )))};

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

Expansion of eta(q^2) * eta(q^4)^2 * eta(q^6)^3 / (eta(q) * eta(q^3) * eta(q^12)^2) in powers of q.
Expansion of (theta_3(q)^2 + 3*theta_3(q^3)^2) / 4 in powers of q.
Euler transform of period 12 sequence [ 1, 0, 2, -2, 1, -2, 1, -2, 2, 0, 1, -2, ...].
Moebius transform is period 12 sequence [ 1, 0, 2, 0, 1, 0, -1, 0, -2, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2-(-1)^e, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) == (1-(-1)^e)/2 if p == 3 (mod 4).
G.f.: 1 + Sum_{k>0} (x^k + x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122857.
a(12*n + 7) = a(12*n + 11) = 0. a(2*n) = a(n). a(2*n + 1) = A138741(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 1) = A008441(n). a(4*n + 3) = 3 * A008441(n). a(6*n + 1) = A002175(n). a(6*n + 5) = 2 * A121444(n). a(8*n + 1) = A113407(n). a(8*n + 3) = 3 * A113407(n). a(8*n + 5) = 2 * A053692(n). a(8*n + 7) = 6 * A053692(n). a(9*n) = A125061(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Nov 24 2023

A138741 Expansion of q^(-1/2) * eta(q)^3 * eta(q^4) * eta(q^12) / (eta(q^2)^2 * eta(q^3)) in powers of q (unsigned).

Original entry on oeis.org

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

Offset: 0

Michael Somos, Mar 27 2008

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

			G.f. = 1 + 3*x + 2*x^2 + x^4 + 2*x^6 + 6*x^7 + 2*x^8 + 3*x^12 + 3*x^13 + ...
G.f. = q + 3*q^3 + 2*q^5 + q^9 + 2*q^13 + 6*q^15 + 2*q^17 + 3*q^25 + 3*q^27 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002175, A008441, A019669, A116604, A121444, A132003.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ x^(-1/2) (EllipticTheta[ 2, 0, x]^2 + 3 EllipticTheta[ 2, 0, x^3]^2) / 4, {x, 0, n}]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 3, 1, # == 3, 2 - (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger[2 n + 1])]; (* Michael Somos, Sep 08 2015 *)
QP = QPochhammer; s = QP[q^2]^7*QP[q^3]*QP[q^12]^2 / (QP[q]^3*QP[q^4]^2* QP[q^6]^3) + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)

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

{a(n) = my(A, p, e); 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==3, 2 - (-1)^e, p%12<6, e+1, 1-e%2 )))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A) * eta(x^12 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A)^3), n))};

Expansion of q^(-1/2) * (theta_2(q)^2 + 3 * theta_2(q^3)^2) / 4 in powers of q.
Expansion of phi(q) * psi(q) * psi(q^3) / phi(q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 12 sequence [ 3, -4, 2, -2, 3, -2, 3, -2, 2, -4, 3, -2, ...].
Moebius transform is period 24 sequence [ 1, -1, 2, 0, 1, -2, -1, 0, -2, -1, -1, 0, 1, 1, 2, 0, 1, 2, -1, 0, -2, 1, -1, 0, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1 + (-1)^e, b(p^e) = e+1 if p == 1, 5 (mod 12), b(p^e) = (1+(-1)^e)/2 if p = 7, 11 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 6 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132003.
a(6*n + 3) = a(6*n + 5) = 0.
a(n) = (-1)^n * A116604(n). a(2*n) = A008441(n).
a(6*n) = A002175(n). a(6*n + 1) = 3 * A008441(n). a(6*n + 2) = 2 * A121444(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Dec 28 2023

A286180 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k in powers of x.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 1, 0, 1, 4, 3, 2, 0, 0, 1, 5, 6, 4, 2, 0, 0, 1, 6, 10, 8, 6, 0, 1, 0, 1, 7, 15, 15, 13, 3, 3, 0, 0, 1, 8, 21, 26, 25, 12, 6, 2, 0, 0, 1, 9, 28, 42, 45, 31, 14, 9, 0, 0, 0, 1, 10, 36, 64, 77, 66, 35, 24, 3, 2, 1, 0, 1, 11, 45

Offset: 0

Seiichi Manyama, May 07 2017

A(n, k) is the number of ways of writing n as the sum of k triangular numbers.

			Square array begins:
   1, 1, 1, 1,  1,  1, ...
   0, 1, 2, 3,  4,  5, ...
   0, 0, 1, 3,  6, 10, ...
   0, 1, 2, 4,  8, 15, ...
   0, 0, 2, 6, 13, 25, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-12 give A000007, A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787.

Main diagonal gives A106337.

Table[Function[k, SeriesCoefficient[Product[(1 + x^i) (1 - x^(2 i)), {i, Infinity}]^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten (* Michael De Vlieger, May 07 2017 *)

G.f. of column k: (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k.

A004020 Theta series of square lattice with respect to edge.

Original entry on oeis.org

2, 4, 2, 4, 4, 0, 6, 4, 0, 4, 4, 4, 2, 4, 0, 4, 8, 0, 4, 0, 2, 8, 4, 0, 4, 4, 0, 4, 4, 4, 2, 8, 0, 0, 4, 0, 8, 4, 4, 4, 0, 0, 6, 4, 0, 4, 8, 0, 4, 4, 0, 8, 0, 0, 0, 8, 6, 4, 4, 0, 4, 4, 0, 0, 4, 4, 8, 4, 0, 4, 4, 0, 6, 4, 0, 0, 8, 0, 4, 4, 0, 12, 0, 4, 4, 0, 0, 4, 4, 0, 2, 8, 4, 4, 8, 0, 0, 4, 0, 4, 4, 4, 4, 0

Offset: 0

N. J. A. Sloane

nonn

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

Number of solutions in integers of n = x^2 + y^2 + y.

			G.f. = 2 + 4*x + 2*x^2 + 4*x^3 + 4*x^4 + 6*x^6 + 4*x^7 + 4*x^9 + 4*x^10 + ...
G.f. = 2*q + 4*q^5 + 2*q^9 + 4*q^13 + 4*q^17 + 6*q^25 + 4*q^29 + 4*q^37 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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. A000122, A000700, A000796, A004531, A008441, A010054, A121373.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x] / x^(1/4), {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)
s = 2*QPochhammer[q^2]^4/QPochhammer[q]^2+O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015 *)

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

{a(n) = 2 * if( n<1, n==0, polcoeff( sum(k=0, (sqrtint(8*n + 1) - 1)\2, x^(k*(k + 1)/2), x*O(x^n))^2, n))};

G.f.: 2 * (Sum_{k>0} x^((k^2 - k)/2))^2 = (Sum_{k in Z} x^(k^2 + k)) * (Sum_{k in Z} x^(k^2)).
Expansion of q^(-1/2) * c(q) / 2 in powers of q^2 where c(q) is the third function in the quadratic Gauss AGM. - Michael Somos, Feb 10 2006
Expansion of 2 * phi(x) * psi(x^2) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 10 2006
a(n) = 2*A008441(n) = A004531(4*n + 1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi (A000796). - Amiram Eldar, Oct 15 2022

A113652 Expansion of (1 - theta_4(q)^2) / 4 in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Nov 03 2005

			G.f. = x - x^2 - x^4 + 2*x^5 - x^8 + x^9 - 2*x^10 + 2*x^13 - x^16 + 2*x^17 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(v).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 576.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28, Article 269.

G. C. Greubel, Table of n, a(n) for n = 1..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A002654, A053692, A008441, A104794, A113407, A122856, A122865, A258277, A258278.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[4, 0, q]^2) / 4, {q, 0, n}]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := With[ {m = InverseEllipticNomeQ @ -q}, SeriesCoefficient[(1 - EllipticK[m] / (Pi/2)) / 4, {q, 0, n}]]; (* Michael Somos, Jun 06 2015 *)

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

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, -1, p%4==1, e+1, !(e%2))))};

{a(n) = if( n<1, 0, direuler(p=2, n, if( p==2, 1 - X/(1 - X), 1 / ((1 - X) * (1 - kronecker( -4, p)*X))) )[n])};

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

a(n) is multiplicative with a(2^e) = -1 if e>0, a(p^e) = e+1 if p == 1 (mod 4), (1 + (-1)^e)/2 if p == 3 (mod 4).
Expansion of (1 - eta(q)^4 / eta(q^2)^2) / 4 in powers of q.
Moebius transform is period 8 sequence [ 1, -2, -1, 0, 1, 2, -1, 0, ...].
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 - 2*u3 + u6 - u1^2 + 3*u3^2 + 2*u1*u3 - 4*u2*u6.
G.f.: Sum_{k>0} -(-1)^k * x^((k^2 + k)/2) / (1 + x^k).
G.f.: Sum_{k>0} -(-1)^k * x^k / (1 + x^(2*k)).
G.f.: Sum_{k>0} -(-1)^k * x^(2*k - 1) / (1 + x^(2*k - 1)).
a(n) = -(-1)^n * A002654(n). a(n) = - A104794(n) / 4 unless n = 0.
a(2*n) = - A002654(n). a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(4*n + 1) = A008441(n). a(4*n + 3) = 0. a(6*n + 2) = - A122856(n). a(6*n + 4 ) = - A122856(n). - Michael Somos, Jun 06 2015
a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). a(9*n + 3) = a(9*n + 6) = 0. - Michael Somos, Jun 06 2015

A116604 Expansion of q^(-1/2) * eta(q)^3 * eta(q^4) * eta(q^12) / (eta(q^2)^2 * eta(q^3)) in powers of q.

Original entry on oeis.org

1, -3, 2, 0, 1, 0, 2, -6, 2, 0, 0, 0, 3, -3, 2, 0, 0, 0, 2, -6, 2, 0, 2, 0, 1, -6, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, -9, 0, 0, 1, 0, 4, -6, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, -6, 2, 0, 2, 0, 1, -6, 4, 0, 0, 0, 0, -6, 2, 0, 0, 0, 4, -3, 2, 0, 2, 0, 2, -6, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, -6, 4, 0, 0, 0, 2, -12, 2, 0, 0, 0, 4, 0, 0

Offset: 0

Michael Somos, Feb 18 2006, Apr 03 2008

sign

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

			1 - 3*x + 2*x^2 + x^4 + 2*x^6 - 6*x^7 + 2*x^8 + 3*x^12 - 3*x^13 + ...
q - 3*q^3 + 2*q^5 + q^9 + 2*q^13 - 6*q^15 + 2*q^17 + 3*q^25 - 3*q^27 + ...

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. A002175, A008441, A121450, A138741 (unsigned version).

QP = QPochhammer; s = QP[q]^3*QP[q^4]*(QP[q^12]/(QP[q^2]^2*QP[q^3])) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -4, n/d) * [ -2, 1, 1][d%3 + 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, -1 + 2 * (-1)^e, if( p%12 < 6, e+1, (1 + (-1)^e) / 2)))))) }

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

G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)) * (1 - x^k + x^(2*k)) * (1 + x^(6*k)).
G.f.: Sum_{k>=0} x^(3*k) / (1 + x^(6*k + 1)) - 2*x^(3*k + 1)  /(1 + x^(6*k+3)) + x^(3*k + 2) / (1 + x^(6*k + 5)).
Expansion of psi(q^2)^2 - 3 * q * psi(q^6)^2 in powers of q where psi() is a Ramanujan theta function.
Euler transform of period 12 sequence [ -3, -1, -2, -2, -3, 0, -3, -2, -2, -1, -3, -2, ...].
Moebius transform is period 24 sequence [ 1, -1, -4, 0, 1, 4, -1, 0, 4, -1, -1, 0, 1, 1, -4, 0, 1, -4, -1, 0, 4, 1, -1, 0, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = -1 + 2 * (-1)^e, b(p^e) = e + 1 if p == 1, 5 (mod 12), b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 12 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A121450.
a(6*n + 3) = a(6*n + 5) = 0. a(6*n) = A002175(n). a(2*n) = A008441(n).

A138949 Expansion of (3 * phi(q^3)^2 - phi(q)^2) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Apr 03 2008

sign

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

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
L.-C. Shen, On the Modular Equations of Degree 3, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1101-1114. See p. 1108, Eq. (3.20).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008441, A116604, A122856, A122865, A138950.

a[ n_] := SeriesCoefficient[ (3 EllipticTheta[ 3, 0, q^3]^2 - EllipticTheta[ 3, 0, q]^2) / 2, {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -2 DivisorSum[ n, KroneckerSymbol[ -4, n/#] {1, 1, -2}[[Mod[#, 3, 1]]] &]]; (* Michael Somos, Sep 07 2015 *)

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

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, -1 + 2 * (-1)^e, p%12 < 6, e+1, 1-e%2))) };

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

Expansion of phi(-q) * phi(-q^2) * chi(q^3) / chi(-q^3) in powers of q where phi(), chi() are Ramanujan theta functions.
Expansion of eta(q)^2 * eta(q^2) * eta(q^6)^3 / (eta(q^3)^2 * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -2, -3, 0, -2, -2, -4, -2, -2, 0, -3, -2, -2, ...].
Moebius transform is period 12 sequence [ -2, 0, 8, 0, -2, 0, 2, 0, -8, 0, 2, 0, ...].
a(n) = -2 * b(n) where b() is multiplicative and b(2^e) = 1, b(3^e) = -1 + 2 * (-1)^e, b(p^e) = e+1 if p == 1, 5 (mod 12), b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113446.
G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 - x^k + x^(2*k))^2 / ((1 + x^(2*k))^2 * (1 - x^(2*k) + x^(4*k))).
G.f.: 1 - 2 * Sum_{k>0} (f(3*k - 2) + f(3*k - 1) - 2 * f(3*k)) where f(n) := x^n / (1 + x^(2*n)).
a(12*n + 7) = a(12*n + 11) = 0. a(2*n) = a(n).
a(n) = -2 * A138950(n) unless n=0. a(2*n + 1) = -2 * A116604(n).
a(3*n + 1) = A122865(n). a(3*n + 2) = -2 * A122856(n). a(4*n + 1) = -2 * A008441(n).

A138950 Expansion of (2 - 3 * phi(q^3)^2 + phi(q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Apr 03 2008

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

			G.f. = q + q^2 - 3*q^3 + q^4 + 2*q^5 - 3*q^6 + q^8 + q^9 + 2*q^10 - 3*q^12 + ...

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

Cf. A008441, A116604, A122856, A122865, A138949.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -4, n/#] {1, 1, -2}[[Mod[#, 3, 1]]] &]]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ (2 - 3 EllipticTheta[ 3, 0, q^3]^2 + EllipticTheta[ 3, 0, q]^2) / 4, {q, 0, n}]; (* Michael Somos, Sep 07 2015 *)

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

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, -1 + 2 * (-1)^e, p%12 < 6, e+1, 1-e%2)))};

Expansion of (1 - eta(q)^2 * eta(q^2) * eta(q^6)^3 / (eta(q^3)^2 * eta(q^4) * eta(q^12))) / 2 in powers of q.
Moebius transform is period 12 sequence [ 1, 0, -4, 0, 1, 0, -1, 0, 4, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = -1 + 2 * (-1)^e, a(p^e) = e+1 if p == 1, 5 (mod 12), a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12).
G.f.: Sum_{k>0} f(3*k - 2) + f(3*k - 1) - 2 * f(3*k) where f(n) := x^n / (1 + x^(2*n)).
a(12*n + 7) = a(12*n + 11) = 0. a(2*n) = a(n). a(2*n + 1) = A116604(n).
-2 * a(n) = A138949(n) unless n=0. a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 1) = A008441(n).

cp's OEIS Frontend

A134343 Expansion of psi(-x)^2 in powers of x where psi() is a Ramanujan theta function.

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A008442 Expansion of Jacobi theta constant (theta_2(2z))^2/4.

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A125061 Expansion of psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) in powers of q where psi(), chi() are Ramanujan theta functions.

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A138741 Expansion of q^(-1/2) * eta(q)^3 * eta(q^4) * eta(q^12) / (eta(q^2)^2 * eta(q^3)) in powers of q (unsigned).

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A286180 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of (Product_{j>0} (1 + x^j) * (1 - x^(2*j)))^k in powers of x.

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A004020 Theta series of square lattice with respect to edge.

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