A097195 -id:A097195 - cp's OEIS frontend

A193514 Expansion of phi(-q)^2 * phi(-q^9) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

1, -4, 4, 2, -4, 0, 4, -8, 4, 2, 0, 0, 2, -8, 8, 0, -4, 0, 4, -8, 0, 4, 0, 0, 4, -4, 8, 2, -8, 0, 0, -8, 4, 0, 0, 0, 2, -8, 8, 4, 0, 0, 8, -8, 0, 0, 0, 0, 2, -12, 4, 0, -8, 0, 4, 0, 8, 4, 0, 0, 0, -8, 8, 4, -4, 0, 0, -8, 0, 0, 0, 0, 4, -8, 8, 2, -8, 0, 8, -8, 0, 2, 0, 0, 4, 0, 8, 0, 0, 0, 0, -16, 0, 4, 0, 0, 4, -8

Offset: 0

Michael Somos, Jul 29 2011

sign

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

			G.f. = 1 - 4*q + 4*q^2 + 2*q^3 - 4*q^4 + 4*q^6 - 8*q^7 + 4*q^8 + 2*q^9 + 2*q^12 + ...

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, A033762, A097195, A112604, A112605.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^9] / EllipticTheta[ 4, 0, q^3], {q, 0, n}];

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

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

Expansion of (-2 * a(q) +2 * a(q^2) +3 * a(q^3)) / 3 = b(q) * (b(q) + 2 * b(q^2)) / (3 * b(q^2)) in powers of q where a(), b() are cubic AGM functions.
Expansion of eta(q)^4 * eta(q^6) * eta(q^9)^2 / (eta(q^2)^2 * eta(q^3)^2 * eta(q^18)) in powers of q.
Euler transform of period 18 sequence [ -4, -2, -2, -2, -4, -1, -4, -2, -4, -2, -4, -1, -4, -2, -2, -2, -4, -2, ...].
Moebius transform is period 18 sequence [ -4, 8, 6, -8, 4, -6, -4, 8, 0, -8, 4, 6, -4, 8, -6, -8, 4, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 432^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A193426.
a(3*n) = A123330(n). a(3*n + 1) = -4 * A033687(n). a(6*n + 1) = -4 * A097195(n). a(6*n + 2) = 4 * A033687(n). a(6*n + 3) = 2 * A033762(n). a(6*n + 4) = 4 * A033687(n). a(8*n + 2) = 4 * A112604(n). a(8*n + 6) = 4 * A112605(n). a(6*n + 5) = 0. a(4*n) = a(n).

A253623 Expansion of phi(q) * f(q, q^2)^2 / f(q^2, q^4) in powers of q where phi(), f() are Ramanujan theta functions.

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, 0, 12, 8, 0, 4, 0, 0

Offset: 0

Michael Somos, Jan 06 2015

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

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

Cf. A004016, A033687, A033762, A097195, A112604, A112605, A121361, A123884, A186706, A244339, A253625.

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

A := Basis( ModularForms( Gamma1(12), 1), 83); A[1] + 4*A[2] + 6*A[3] + 4*A[4];

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

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

{a(n) = if( n<1, n==0, 2 * (-1)^n * 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^2 + A)^8 * eta(x^3 + A)^4 * eta(x^12 + A) / (eta(x + A)^4 * eta(x^4 + A)^3 * eta(x^6 + A)^4), n))};

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

Expansion of phi(q)^2 * phi(-q^3)^2 / (phi(-q^2) * phi(-q^6)) = psi(q) * psi(-q^3) * (chi(q) * chi(-q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of (2*a(q) + 3*a(q^2) - 2*a(q^4)) / 3 = (b(q) - 2*b(q^4)) * (b(q) - 4*b(q^4)) / (3*b(q^2)) in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q^2)^8 * eta(q^3)^4 * eta(q^12) / (eta(q)^4 * eta(q^4)^3 * eta(q^6)^4) in powers of q.
Euler transform of period 12 sequence [ 4, -4, 0, -1, 4, -4, 4, -1, 0, -4, 4, -2, ...].
Moebius transform is period 12 sequence [ 4, 2, 0, -6, -4, 0, 4, 6, 0, -2, -4, 0, ...].
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 A253625.
a(n) = 4*b(n) where b() is multiplicative with b(2^e) = (3/4) * (1 - (-1)^e) 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).
G.f.: 1 + Sum_{k>0} (1 + (k mod 2)) * q^k / (1 - q^k + q^(2*k)).
G.f.: Product_{k>0} (1 + q^k) * (1 - q^(2*k)) * (1 - q^(3*k)) * (1 + q^(6*k)) / ((1 + q^(2*k)) * (1 - q^k + q^(2*k)))^3.
a(n) = (-1)^n * A244339(n). 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 + 2) = 6 * A033687(n). a(6*n + 4) = a(6*n = 5) = 0.
a(12*n + 1) = 4 * A123884(n). a(12*n + 2) = 6 * A097195(n). a(12*n + 3) = 4 * A112604(n). a(12*n + 7) = 8 * A121361(n). a(12*n + 9) = 4 * A112605(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(3) = 3.627598... (A186706). - Amiram Eldar, Dec 30 2023

A260945 Expansion of (2*b(q^4) - b(q) - b(q^2)) / 3 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

0, 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

Offset: 0

Michael Somos, Aug 04 2015

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

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. A033687, A033762, A097195, A093829, A112848, A113447, A123530, A123863, A227696, A246838, A248897, A253243, A260941, A260942, A260943, A260944.

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

A := Basis( ModularForms( Gamma1(36), 1), 80); A[2] + A[3] - 2*A[4] - A[5] - 2*A[7] + 2*A[8] + A[9] - 2*A[10] + 2*A[13] + 2*A[14] + 2*A[15] - A[17] - 2*A[19] - 4*A[20];

a[ n_] := If[ n < 1, 0, Sum[ {1, 1, 0, -1, -1, 0}[[Mod[ d, 6, 1]]] {1, 0, -2, 0, 1, 0}[[Mod[ n/d, 6, 1]]], {d, Divisors @ n}]]
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # == 2, -(-1)^#2, # == 3, -2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)] EllipticTheta[ 3, 0, q] / (2 q^(1/4) QPochhammer[ q^6]), {q, 0, n}];

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

{a(n) = if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(-1)^e, p==3, -2, p%6==5, 1-e%2, e+1)))};

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

Expansion of (a(q) + a(q^2) - 3*a(q^3) - 2*a(q^4) - 3*a(q^6) + 6*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.
Expansion of q * phi(q) * psi(-q) * psi(-q^9) / f(-q^6) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Expansion of eta(q^2)^4 * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^6) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 1, -3, 1, -2, 1, -2, 1, -2, 0, -3, 1, -1, 1, -3, 1, -2, 1, -2, 1, -2, 1, -3, 1, -1, 1, -3, 0, -2, 1, -2, 1, -2, 1, -3, 1, -2, ...].
Moebius transform is period 36 sequence [ 1, 0, -3, -2, -1, 0, 1, 2, 0, 0, -1, 6, 1, 0, 3, -2, -1, 0, 1, 2, -3, 0, -1, -6, 1, 0, 0, -2, -1, 0, 1, 2, 3, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -(-1)^e if e>0, 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. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123863.
a(2*n) = A112848(n). a(2*n + 1) = A123530(n). a(3*n) = -2 * A113447(n). a(3*n + 1) = A227696(n).
a(4*n) = - A112848(n). a(4*n + 1) = A253243(n). a(4*n + 2) = A123530(n). a(4*n + 3) = -2 * A246838(n).
a(6*n) = -2 * A093829(n). a(6*n + 1) = A097195(n). a(6*n + 2) = A033687(n). a(6*n + 3) = -2 * A033762(n). a(6*n + 5) = 0.
a(8*n + 1) = A260941(n). a(8*n + 2) = A253243(n). a(8*n + 3) = -2 * A260943(n). a(8*n + 4) = - A123530(n). a(8*n + 5) = 2 * A260942(n). a(8*n + 6) = -2 * A246838(n). a(8*n + 7) = 2 * A260944(n).
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 23 2024

A260958 Expansion of (a(q) - 3a(q^2) + 3a(q^3) - 4a(q^4) + 3a(q^6)) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 05 2015

sign

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. = x - 3*x^2 + 4*x^3 - 3*x^4 + 2*x^7 - 3*x^8 + 4*x^9 + 2*x^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. A033687, A033762, A097195, A113448, A122861.

a[ n_] := If[ n < 1, 0, {1, -3, 4, -3, 1, 0}[[Mod[ n, 6, 1]]] Sum[ {1, 0, 0, 0, -1, 0}[[Mod[ d, 6, 1]]], {d, Divisors @ n}]];
a[ n_] := If[ n < 1 || Mod[n, 6] == 0, 0, Times @@ (Which[ # == 1, 1, # == 2, -2 - Mod[#2, 2], # == 3, 4, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)];
a[ n_] := SeriesCoefficient[ QPochhammer[ x] EllipticTheta[ 3, 0, x^3] QPochhammer[ -x^3, x^6]^2 EllipticTheta[ 2, 0, x^(9/2)] / (2 x^(1/8) QPochhammer[-x, x^2]^2 EllipticTheta[ 3, 0, x^9]), {x, 0, n}];

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

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

Expansion of q * f(q) * phi(q^3) * chi(q^3)^2 * psi(q^9) / (chi(q)^2 * phi(q^9)) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of eta(q)^3 * eta(q^4)^2 * eta(q^6)^9 * eta(q^9) * eta(q^36)^2 / (eta(q^2)^4 * eta(q^3)^4 * eta(q^12)^4 * eta(q^18)^3) in powers of q.
Euler transform of period 36 sequence [ -3, 1, 1, -1, -3, -4, -3, -1, 0, 1, -3, -2, -3, 1, 1, -1, -3, -2, -3, -1, 1, 1, -3, -2, -3, 1, 0, -1, -3, -4, -3, -1, 1, 1, -3, -2, ...].
Moebius transform is period 36 sequence [ 1, -4, 3, 0, -1, 0, 1, 0, 0, 4, -1, 0, 1, -4, -3, 0, -1, 0, 1, 0, 3, 4, -1, 0, 1, -4, 0, 0, -1, 0, 1, 0, -3, 4, -1, 0, ...].
a(2*n) = A113448(n). a(3*n + 1) = A122861(n). a(6*n) = 0. a(6*n + 1) = A097195(n). a(6*n + 2) = a(12*n + 4) = -3 * A033687(n). a(6*n + 3) = 4 * A033762(n). a(6*n + 5) = a(12*n + 10) = 0.

A261884 Expansion of (a(q) - a(q^2) - 2a(q^3) + 2a(q^6)) / 6 in powers of q where a() is a cubic AGM function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 04 2015

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

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

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

Cf. A033687, A033762, A093829, A097195, A035178 (apparently gives the absolute values).

Cf. A004016, A005882, A005928.

A004016[q_] := (QPochhammer[q]^3 + 9*q*QPochhammer[q^9]^3)/ QPochhammer[q^3]; A261884[n_] := SeriesCoefficient[(A004016[q] - A004016[q^2] - 2*A004016[q^3] + 2*A004016[q^6])/6, {q, 0, n}]; Table[A261884[n], {n, 1, 50}] (* G. C. Greubel, Sep 24 2017 *)

{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)^e, p==3, -1, p%6==1, e+1, 1-e%2)))};

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

Moebius transform is period 18 sequence [ 1, -2, -2, 2, -1, 4, 1, -2, 0, 2, -1, -4, 1, -2, 2, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = -1 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} F(x^(6*k - 5)) - F(x^(6*k - 3)) + F(x^(6*k - 1)) where F(x) := x / (1 + x + x^2).
a(n) = A093829(n) unless n == 0 (mod 3). a(2*n) = - a(n). a(3*n + 1) = A033687(n).
a(6*n + 1) = A097195(n). a(6*n + 3) = - A033762(n). a(6*n + 5) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(18*sqrt(3)) = 0.100766631346... . - Amiram Eldar, Nov 23 2023

cp's OEIS Frontend

A193514 Expansion of phi(-q)^2 * phi(-q^9) / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

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A253623 Expansion of phi(q) * f(q, q^2)^2 / f(q^2, q^4) in powers of q where phi(), f() are Ramanujan theta functions.

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A260945 Expansion of (2*b(q^4) - b(q) - b(q^2)) / 3 in powers of q where b() is a cubic AGM theta function.

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A260958 Expansion of (a(q) - 3*a(q^2) + 3*a(q^3) - 4*a(q^4) + 3*a(q^6)) / 6 in powers of q where a() is a cubic AGM theta function.

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Formula

A261884 Expansion of (a(q) - a(q^2) - 2*a(q^3) + 2*a(q^6)) / 6 in powers of q where a() is a cubic AGM function.

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PARI

PARI

Formula

A260958 Expansion of (a(q) - 3a(q^2) + 3a(q^3) - 4a(q^4) + 3a(q^6)) / 6 in powers of q where a() is a cubic AGM theta function.

A261884 Expansion of (a(q) - a(q^2) - 2a(q^3) + 2a(q^6)) / 6 in powers of q where a() is a cubic AGM function.