A122865 -id:A122865 - cp's OEIS frontend

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

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

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).

A163746 Expansion of (theta_3(q)^2 + 3 * theta_3(q^3)^2) / 4 - 1 in powers of q.

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

Offset: 1

Michael Somos, Aug 03 2009

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 + ...

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 = 1..5000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A019669, A122856, A122865, A125061, A138741.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[# < 3, 1, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger @ n)]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^2 + 3 EllipticTheta[ 3, 0, q^3]^2) / 4 - 1, {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)

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

{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, e%2*2 + 1, p%4==1, e+1, 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) - 1, n))};

Expansion of psi(q) * psi(q^2) * chi(q^3) * chi(-q^6) - 1 in powers of q where psi(), chi() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^4)^2 * eta(q^6)^3 / (eta(q) * eta(q^3) * eta(q^12)^2) - 1 in powers of q.
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). [corrected by Amiram Eldar, Nov 14 2023]
G.f.: Sum_{k>0} (x^k + x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
a(n) = A125061(n) unless n=0. 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). - Michael Somos, Sep 02 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Nov 14 2023

A258256 Expansion of f(q^3) * psi(-q^3)^3 / (psi(-q) * psi(-q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 0, 0, 1, 4, 2, 0, 0, 2, 0, 0, 1, 2, 4, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 4, 2, 0, 0, 2, 2, 0, 0, 0, 8, 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, 4, 2, 2, 0, 0, 0, 0, 0, 2, 4, 2, 0, 0, 4, 0

Offset: 0

Michael Somos, May 24 2015

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

			G.f. = 1 + q + q^2 + q^4 + 2*q^5 + q^8 + 4*q^9 + 2*q^10 + 2*q^13 + q^16 + ...

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, A019670, A089801, A121444, A122856, A122856, A256282, A258260.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + A[2] + A[3] + A[5] + 2*A[6] + A[9] + 4*A[10] + 2*A[11] + 2*A[14] + A[17] + 2*A[18] + 4*A[19];

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, {1, 2, -1, 0}[[Mod[#, 4, 1]]] If[ Divisible[ #, 9], 4, 1] (-1)^(Boole[Mod[#, 8] == 6] + n + #) &]];
a[ n_] := If[ n < 2, Boole[n >= 0], Times @@ (Which[ # == 2, 1, Mod[#, 4] == 1, #2 + 1, True, If[# == 3, 4, 1] Mod[#2 + 1, 2]] & @@@ FactorInteger[n])];
a[ n_] := SeriesCoefficient[ q^(1/8) QPochhammer[ -q^3] EllipticTheta[ 2, Pi/4, q^(3/2)]^3 / (Sqrt[2] EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)]), {q, 0, n}];

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

{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%4==1, e+1, if( p==3, 4, 1) * (1 - e%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^18 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};

Expansion of eta(q^2) * eta(q^3)^2 * eta(q^12)^2 * eta(q^18) / (eta(q) * eta(q^4) * eta(q^9) * eta(q^36)) in powers of q.
Euler transform of period 36 sequence [1, 0, -1, 1, 1, -2, 1, 1, 0, 0, 1, -3, 1, 0, -1, 1, 1, -2, 1, 1, -1, 0, 1, -3, 1, 0, 0, 1, 1, -2, 1, 1, -1, 0, 1, -2, ...].
Moebius transform is period 36 sequence [1, 0, -1, 0, 1, 0, -1, 0, 4, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -4, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).
a(2*n) = a(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(4*n + 3) = 0. a(12*n + 1) = A002175. a(12*n + 5) = 2 * A121444(n).
a(n) = Sum_{d|n} A258260(d) * (-1)^(n+d) if n>0.
a(n) = (-1)^n * A256282(n). - Michael Somos, Jun 06 2015
a(n) is multiplicative with a(0) = 1, a(2^e) = 1, a(3^e) = 2*(1 + (-1)^e), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Jun 06 2015
Expansion of A0(x)^2 + A0(x)*A1(x) + A1(x)^2 in powers of x where A0(x) = phi(x^9), A1(x) = x * f(x^3, x^15) = x * A089801(x^3). - Michael Somos, Jun 23 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 (A019670). - Amiram Eldar, Nov 24 2023

A122864 Expansion of eta(q^3)^2 * eta(q^4) * eta(q^6)^2 * eta(q^36) / (eta(q) * eta(q^9) * eta(q^12)^2) in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 15 2006

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

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

Cf. A002564, A019670, A035154, A113446, A122856, A122865.

eta[x_] := x^(1/24)*QPochhammer[x]; A122864[n_] := SeriesCoefficient[ (eta[q^3]^2*eta[q^4]*eta[q^6]^2*eta[q^36])/(eta[q]*eta[q^9]*eta[q^12]^2), {q, 0, n}]; Table[A122864[n], {n, 50}] (* G. C. Greubel, Sep 16 2017 *)

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^36 + A) /(eta(x + A) * eta(x^9 + A) * eta(x^12 + A)^2), n))}

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

Euler transform of period 36 sequence [ 1, 1, -1, 0, 1, -3, 1, 0, 0, 1, 1, -2, 1, 1, -1, 0, 1, -2, 1, 0, -1, 1, 1, -2, 1, 1, 0, 0, 1, -3, 1, 0, -1, 1, 1, -2, ...].
Moebius transform is period 36 sequence [ 1, 0, 1, 0, 1, 0, -1, 0, -4, 0, -1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 0, 4, 0, 1, 0, -1, 0, -1, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2*(-1)^(e+1) if e>0, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
a(3*n) = 2 * A113446(n). a(3*n + 1) = A002564(3*n + 1) = A035154(3*n + 1) = A113446(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Oct 15 2022

A129448 Expansion of q * psi(-q) * chi(q^3)^2 * psi(-q^9) in powers of q where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Apr 16 2007

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

Number 50 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - q^2 + q^4 - 2*q^5 - q^8 + 2*q^10 + 2*q^13 + q^16 - 2*q^17 - 2*q^20 + ...

G. C. Greubel, Table of n, a(n) for n = 1..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. A002175, A091400, A121363, A121444, A122856, A122865.

A := Basis( ModularForms( Gamma1(36), 1), 82); A[2] - A[3] + A[5] - 2*A[6] - A[9] + 2*A[11] + 2*A[14] + A[17] - 2*A[18]; /* Michael Somos, Jul 09 2015 */

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ 12, d] KroneckerSymbol[ -3, n/d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^3, q^6]^2 EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)] / (2 q^(5/4)), {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( 12, d) * kronecker( -3, n/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==3, 0, p==2, (-1)^e, p%12>6, !(e%2), (-1)^(e * (p%12==5)) * (e+1))))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^4 * eta(x^9 + A) * eta(x^36 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^18 + A)), n))};

Expansion of eta(q) * eta(q^4) * eta(q^6)^4 * eta(q^9) * eta(q^36) / (eta(q^2) * eta(q^3)^2 * eta(q^12)^2 * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ -1, 0, 1, -1, -1, -2, -1, -1, 0, 0, -1, -1, -1, 0, 1, -1, -1, -2, -1, -1, 1, 0, -1, -1, -1, 0, 0, -1, -1, -2, -1, -1, 1, 0, -1, -2, ...].
a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = 0^e, a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12), a(p^e) = e+1 if p == 1 (mod 12), a(p^e) = (-1)^e * (e+1) if p == 5 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker(12, k) * x^k/ (1 + x^k + x^(2*k)).
|a(n)| = A091400(n). a(3*n) = a(4*n + 3) = 0. a(2*n) = -a(n). a(3*n + 1) = A122865(n). a(3*n + 2) = - A122856(n). a(4*n + 1) = A121363(n). a(12*n + 1) = A002175(n). a(12*n + 5) = -2 * A121444(n).

A256269 Expansion of psi(-q) * chi(q^3) * phi(q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 0, 0, 0, 4, -2, 0, 0, -2, 0, 0, -1, 0, 4, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 4, -2, 0, 0, -2, 0, 0, 0, 0, 8, 0, 0, 0, -1, 0, 0, -2, 0, 0, 0, 0, 0, -2, 0, 0, -2, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 4, -2, 0, 0, 0, 0, 0, 0, 0, 4

Offset: 0

Michael Somos, Jun 01 2015

sign

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

			G.f. = 1 - q - q^4 + 4*q^9 - 2*q^10 - 2*q^13 - q^16 + 4*q^18 - 3*q^25 + ...

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. A122856, A122865.

A := Basis( ModularForms( Gamma1(36), 1), 82); A[1] - A[2] - A[5] + 4*A[10] - 2*A[11] - 2*A[14] - A[17] + 4*A[19];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] / (2^(1/2) q^(1/8)) QPochhammer[ -q^3, q^6] EllipticTheta[ 3, 0, q^9], {q, 0, n}];

{a(n) = if( n<1, n==0, (-1)^(n%3) * (n%3<2) * sumdiv(n, d, [ 0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^18 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) * eta(x^9 + A)^2 * eta(x^36 + A)^2), n))};

Expansion of eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^18)^5 / (eta(q^2) * eta(q^3) * eta(q^12) * eta(q^9)^2 * eta(q^36)^2) in powers of q.
Euler transform of period 36 sequence [ -1, 0, 0, -1, -1, -1, -1, -1, 2, 0, -1, -1, -1, 0, 0, -1, -1, -4, -1, -1, 0, 0, -1, -1, -1, 0, 2, -1, -1, -1, -1, -1, 0, 0, -1, -2, ...].
a(3*n + 1) = - A122865(n). a(6*n + 4) = - A122856(n). a(3*n + 2) = a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.

A258292 Expansion of psi(-q)^2 * chi(q^3)^2 in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 1, 0, -2, 2, 0, 0, 1, 4, -4, 0, 0, -4, 0, 0, -2, 2, 4, 0, 2, 0, 0, 0, 0, -6, 2, 0, 0, 2, 0, 0, 1, 0, -4, 0, 4, -4, 0, 0, -4, 2, 0, 0, 0, 8, 0, 0, 0, -2, 3, 0, -4, 2, 0, 0, 0, 0, -4, 0, 0, -4, 0, 0, -2, 4, 0, 0, 2, 0, 0, 0, 4, -4, 2, 0, 0, 0, 0, 0, 2, 4

Offset: 0

Michael Somos, May 25 2015

sign

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

			G.f. = 1 - 2*q + q^2 - 2*q^4 + 2*q^5 + q^8 + 4*q^9 - 4*q^10 - 4*q^13 + ...kkj

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, A004018, A089807, A122856, A122865, A258228, A258279.

A := Basis( ModularForms( Gamma1(36), 1), 82); A[1] - 2*A[2] + A[3] - 2*A[5] + 2*A[6] + A[9] + 4*A[10] - 4*A[11] - 4*A[14] - 2*A[17] + 2*A[18
] + 4*A[19];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/3, q]^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ q^(1/8) QPochhammer[ -q^3] EllipticTheta[ 2, Pi/4, q^(1/2)]^2 / (Sqrt[2] EllipticTheta[ 2, Pi/4, q^(3/2)]), {q, 0, n}];

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

Expansion of f(q) * psi(-q)^2 / psi(-q^3) in powers of q where psi(), f() are Ramanujan theta functions.
Expansion of f(x*w, x/w)^2 in powers of x where w is a primitive cube root of unity and f() is Ramanujan's general theta function.
Expansion of (eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)))^2 in powers of q.
Euler transform of period 12 sequence [ -2, 0, 0, -2, -2, -2, -2, -2, 0, 0, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 18 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122856.
G.f.: (Product_{k>0} (1 + x^k) * (1 + x^(3*k)) / (1 - x^(2*k) + x^(4*k)))^2.
a(n) = (-1)^n * A258279(n).  Convolution square of A089807.
a(2*n) = A258228(n). a(3*n + 1) = -2 * A122865(n). a(3*n + 2) = A122856(n). a(4*n) = a(n). a(4*n + 3) = 0. a(12*n + 1) = -2 * A002175(n).
a(18*n) = A004018(n). a(18*n + 3) = a(18*n + 6) = a(18*n + 12) = 0.

A281451 Expansion of x * f(x, x) * f(x, x^17) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jan 23 2017

nonn

			G.f. = x + 3*x^2 + 2*x^3 + 2*x^5 + 2*x^6 + 2*x^10 + 2*x^11 + 2*x^17 + ...
G.f. = q^16 + 3*q^25 + 2*q^34 + 2*q^52 + 2*q^61 + 2*q^97 + 2*q^106 + ...

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. A002654, A004018, A019670, A035154, A105673, A113446, A122856, A122857, A122864, A122865.
Cf. A125061, A129448, A138949, A138950, A163746, A256269, A256276, A256280, A258034.
Cf. A258228, A258256, A258278, A258292, A259761.
Cf. A281452, A281453, A281490, A281491, A281492.

a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 7, KroneckerSymbol[ -4, #] &]];
a[ n_] := If[ n < 0, 0, Times @@ (Which[# < 3, 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 7])];
a[ n_] := SeriesCoefficient[ x EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^18] QPochhammer[ -x^17, x^18] QPochhammer[ x^18], {x, 0, n}];

{a(n) = if( n<0, 0, sumdiv(9*n + 7, d, (d%4==1) - (d%4==3)))};

{a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 7); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%4==1, e+1, 1-e%2)))};

{a(n) = if( n<0, 0, my(m = 9*n + 7, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 4 || k%9 == 5), s+=(j>0)+1)); s)};

f(x,x^m) = 1 + Sum_{k>=1} x^((m+1)*k*(k-1)/2) (x^k + x^(m*k)). - N. J. A. Sloane, Jan 30 2017
Euler transform of a period 36 sequence.
G.f.: x * (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 8*k)).
G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(18*k-17)) * (1 + x^(18*k-1)) * (1 - x^(18*k)).
a(4*n) = a(8*n + 7) = a(16*n + 13) = a(32*n + 9) = a(49*n + 7) = a(98*n + 14) = 0.
a(4*n + 1) = A281452(n).  a(8*n + 3) = 2 * A281491(n).  A(16*n + 1) = A281453(n).
a(32*n + 25) = 2 * A281490(n). a(64*n + 49) = a(n). a(128*n + 17) = 2 * A281492(n).
a(n) = A122865(3*n + 2). a(n) = A122856(6*n + 4) = A258278(6*n + 4).
a(n) = b(9*n + 7) where b = A002654, A035154, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
2 * a(n) = b(9*n + 7) where b = A105673, A122857, A258034, A259761.  -2 * a(n) = b(9*n + 7) where b = A138949, A256280, A258292.
a(n) = - A256269(9*n + 7). 4 * a(n) = A004018(9*n + 7).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Jan 20 2025

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

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

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

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A163746 Expansion of (theta_3(q)^2 + 3 * theta_3(q^3)^2) / 4 - 1 in powers of q.

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A258256 Expansion of f(q^3) * psi(-q^3)^3 / (psi(-q) * psi(-q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

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A122864 Expansion of eta(q^3)^2 * eta(q^4) * eta(q^6)^2 * eta(q^36) / (eta(q) * eta(q^9) * eta(q^12)^2) in powers of q.

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