A244375 -id:A244375 - cp's OEIS frontend

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

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

A136748 Expansion of (a(q) - a(q^2) - 4a(q^4) + 4a(q^8)) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jan 22 2008

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

			G.f. = q - q^2 + q^3 - 3*q^4 - q^6 + 2*q^7 + 3*q^8 + q^9 - 3*q^12 + 2*q^13 + ...

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. A033762, A093829, A097195, A112604, A112605, A123484.

Cf. A004016, A005882, A005928.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, (Mod[#, 2] - 4 Boole[Mod[#, 8] == 4]) KroneckerSymbol[ -3, n/#] &]]; (* Michael Somos, Oct 12 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[# == 1 || # == 3, 1, # == 2, If[#2 < 2, -1, -3 (-1)^#2], Mod[#, 6] == 1, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger@n)]; (* Michael Somos, Oct 12 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3+A) * eta(x^4 + A)^4 * eta(x^24 + A)^2 / ( eta(x^2 + A) * eta(x^8+A) * eta(x^12+A) )^2, n))};

{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, if( e<2, -1, -3 * (-1)^e), p==3, 1, p%6>1, !(e%2), e+1)))};

{a(n) = if( n<1, 0, sumdiv(n, d, ((d%2) -4 * (d%8==4)) * kronecker(-3, n/d)))};

Expansion of eta(q) * eta(q^3) * eta(q^4)^4 * eta(q^24)^2 / (eta(q^2) * eta(q^8) * eta(q^12))^2 in powers of q.
Euler transform of period 24 sequence [ -1, 1, -2, -3, -1, 0, -1, -1, -2, 1, -1, -2, -1, 1, -2, -1, -1, 0, -1, -3, -2, 1, -1, -2, ...].
a(n) is multiplicative with a(2) = -1, a(2^e) = -3 * (-1)^e if e>1, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123484.
G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k))^2 * (1 + x^k + x^(2*k)) * (1 - x^(4*k) + x^(8*k))^2.
Moebius transform is period 24 sequence [ 1, -2, 0, -2, -1, 0, 1, 6, 2, -1, 0, 1, -2, 0, -6, -1, 0, 1, 2, 0, 2, -1, 0, ...].
a(2*n) = A244375(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A122861(n).
a(4*n) = -3 * A093829(n). a(4*n + 1) = A112604(n). a(4*n + 2) = -A033762(n). a(4*n + 3) = A112605(n).
a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
Expansion of q * f(-q, -q) * f(q^2, q^10) / f(-q, -q^5)^2 in powers of q where f(, ) is Ramanujan's general theta function. - Michael Somos, Oct 12 2015
Sum_{k=1..n} abs(a(k)) ~ (Pi*sqrt(3)/4) * n. - Amiram Eldar, Jan 28 2024

A244339 Expansion of (-2 * a(q) + 3a(q^2) + 2a(q^4)) / 3 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jun 26 2014

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 - 4*q + 6*q^2 - 4*q^3 + 6*q^6 - 8*q^7 + 6*q^8 - 4*q^9 - 8*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A033762, A097195, A121373, A186706, A244375.

a[ n_] := If[ n < 1, Boole[n == 0], 2 Sum[ (-1)^(n/d) {2, -1, 0, 1, -2, 0}[[ Mod[ d, 6, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^4 QPochhammer[ q^4] QPochhammer[ q^6]^8 / (QPochhammer[ q^2]^4 QPochhammer[ q^3]^4 QPochhammer[ q^12]^3), {q, 0, n}];

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A) * eta(x^6 + A)^8 / (eta(x^2 + A)^4 * eta(x^3 + A)^4 * eta(x^12 + A)^3), n))};

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, n, x^k / (1 + x^k) * [0, -2, 1, 0, -1, 2][k%6 + 1], x * O(x^n)), n))};

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, n, x^k / (1 + x^k + x^(2*k)) * [3, -2, 1, -2][k%4 + 1], x * O(x^n)), n))};

{a(n) = my(A);  if( n<1, n==0, A = factor(n); -4 * prod( j=1, matsize(A)[1], if( p = A[j,1], e = A[j,2]; if( p==2, (1 - (-1)^e) * -3/4, if( p==3, 1, if( p%6 == 1, e+1, (1 + (-1)^e) / 2))))))};

Expansion of b(q) * (b(q) + 2*b(q^4)) / (3 * b(q^2)) in powers of q where b() is a cubic AGM theta function.
Expansion of psi(-q) * chi(-q)^3 * phi(q^3) * chi(q^3)^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q)^4 * eta(q^4) * eta(q^6)^8 / (eta(q^2)^4 * eta(q^3)^4 * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [ -4, 0, 0, -1, -4, -4, -4, -1, 0, 0, -4, -2, ...].
Moebius transform is period 12 sequence [ -4, 10, 0, -6, 4, 0, -4, 6, 0, -10, 4, 0, ...].
a(n) = -4 * b(n) where b(n) is multiplicative with b(2^e) = (1 - (-1)^e) * -3/4 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6) with a(0) = 1.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 48^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A244375.
a(2*n) = A004016(n). a(2*n + 1) = -4 * A033762(n). a(3*n) = a(n). a(6*n + 1) = -4 * A097195(n). a(6*n + 5) = 0.
Sum_{k=1..n} abs(a(k)) ~ (2*Pi/sqrt(3)) * n. - Amiram Eldar, Jun 08 2025

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

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A136748 Expansion of (a(q) - a(q^2) - 4*a(q^4) + 4*a(q^8)) / 6 in powers of q where a() is a cubic AGM theta function.

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A244339 Expansion of (-2 * a(q) + 3*a(q^2) + 2*a(q^4)) / 3 in powers of q where a() is a cubic AGM theta function.

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

A136748 Expansion of (a(q) - a(q^2) - 4a(q^4) + 4a(q^8)) / 6 in powers of q where a() is a cubic AGM theta function.

A244339 Expansion of (-2 * a(q) + 3a(q^2) + 2a(q^4)) / 3 in powers of q where a() is a cubic AGM theta function.