A131943 -id:A131943 - cp's OEIS frontend

A186100 Expansion of 2 * a(q^2)^2 - a(q)^2 in powers of q where a() is a cubic AGM theta function.

1, -12, -12, -12, -12, -72, -12, -96, -12, -12, -72, -144, -12, -168, -96, -72, -12, -216, -12, -240, -72, -96, -144, -288, -12, -372, -168, -12, -96, -360, -72, -384, -12, -144, -216, -576, -12, -456, -240, -168, -72, -504, -96, -528, -144, -72, -288

Offset: 0

Michael Somos, Feb 12 2011

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).
Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 - 12*q - 12*q^2 - 12*q^3 - 12*q^4 - 72*q^5 - 12*q^6 - 96*q^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A008653, A098098, A111932, A121443, A131943, A131946, A186099.

a[ n_] := If[ n < 1, Boole[n == 0], -12 DivisorSum[ n, # Boole[ 1 == GCD[#, 6]] &]]; (* Michael Somos, Jul 07 2015 *)
a[ n_] := SeriesCoefficient[(EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^3])^2 - 1/2 (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)])^2, {x, 0, n}]; (* Michael Somos, Jul 07 2015 *)

{a(n) = if( n<1, n==0, -12 * sumdiv( n, d, d * (1 == gcd( d, 6) ) ) )};

{a(n) = if( n<1, n==0, -12 * direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n])};

Expansion of b(x) * b(x^2) - c(x) * c(x^2) in powers of x where b(), c() are cubic AGM functions.
Expansion of (phi(-x) * phi(-x^3))^2 - 8 * x * (psi(x) * psi(x^3))^2 in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of (P(q) - 2*P(q^2) - 3*P(q^3) + 6*P(q^6)) / 2 in powers of q where P() is a Ramanujan Eisenstein series. - Michael Somos, Jul 07 2015
a(n) = -12 * A186099(n) if n>0. a(2*n) = a(n). a(2*n + 1) = - A008653(2*n + 1). a(n) = 2 * A008653(n) - A008653(2*n) = A131946(n) - 8 * A111932(n) = A131943(n) - 9 * A121443(n).
a(3*n) = a(n). a(6*n + 5) = -72 * A098098(n).- Michael Somos, Jul 07 2015

A132000 Expansion of (1/3) * b(q) * b(q^2) * c(q)^2 / c(q^2) in powers of q where b(), c() are cubic AGM functions.

Original entry on oeis.org

1, -1, -5, -1, 11, 24, -5, -50, -53, -1, 120, 120, 11, -170, -250, 24, 203, 288, -5, -362, -264, -50, 600, 528, -53, -601, -850, -1, 550, 840, 120, -962, -821, 120, 1440, 1200, 11, -1370, -1810, -170, 1272, 1680, -250, -1850, -1320, 24, 2640, 2208, 203, -2451

Offset: 0

Michael Somos, Aug 06 2007

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. = 1 - x - 5*x^2 - x^3 + 11*x^4 + 24*x^5 - 5*x^6 - 50*x^7 - 53*x^8 - x^9 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.71).

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

Cf. A113261, A122373, A123330, A131943.

A := Basis( ModularForms( Gamma1(6), 3), 50); A[1] - A[2] - 5*A[3] - A[4]; /* Michael Somos, Nov 03 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, #^2 (-1)^# KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Nov 03 2015 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 4, 0, q]^2 EllipticTheta[ 4, 0, q^3]^2 EllipticTheta[ 2, 0, q^(1/2)]^3 / EllipticTheta[ 2, 0, q^(3/2)], {q, 0, n}]; (* Michael Somos, Nov 03 2015 *)
a[ n_] := SeriesCoefficient[(9 EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3]^5 - EllipticTheta[ 4, 0, q]^5 EllipticTheta[ 4, 0, q^3]) / 8, {q, 0, n}]; (* Michael Somos, Nov 03 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^2]^4 QPochhammer[ q^3]^5 / QPochhammer[ q^6]^4, {q, 0, n}]; (* Michael Somos, Nov 03 2015 *)

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

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

{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==3, 1, p==2, 2 + ((-4)^(e+1) - 1) / 5, p = p^2 * kronecker(-3, p); (p^(e+1) - 1) / (p-1) )))};

Expansion of phi(-q)^2 * phi(-q^3)^2 * psi(q)^3 / psi(q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^2)^4 * eta(q^3)^5 / eta(q^6)^4 in powers of q.
Euler transform of period 6 sequence [-1, -5, -6, -5, -1, -6, ...].
a(n) = -b(n) where b() is multiplicative with b(2^e) = 2+((-4)^(e+1)-1)/5, b(3^e) = 1, b(p^e) = (q^(e+1) - 1) / (q-1) where q = p^2*Kronecker(-3, p) if p > 3.
a(3*n) = a(n).
G.f.: 1 - Sum_{k>0} k^2 * Kronecker(-3, k) * x^k / (1 - (-x)^k) = Product_{k>0} (1  - x^(3*k)) * (1 - x^k)^5 / (1 - x^k + x^(2*k))^4.
a(n) = (-1)^n * A113261(n). Convolution of A123330 and A131943.
a(n) = -A132000(n) unless n=0.
Expansion of (9 * phi(-q) * phi(-q^3)^5 - phi(-q)^5 * phi(-q^3)) / 8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Nov 03 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 15552^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A122373. - Michael Somos, Nov 03 2015

A131944 Expansion of (1 - b(q)b(q^2)) / 3 where b() is a cubic AGM function. Expansion of (1 - eta(q)^3 eta(q^2)^3 / (eta(q^3) * eta(q^6))) / 3 in powers of q.

Original entry on oeis.org

1, 1, -5, 1, 6, -5, 8, 1, -23, 6, 12, -5, 14, 8, -30, 1, 18, -23, 20, 6, -40, 12, 24, -5, 31, 14, -77, 8, 30, -30, 32, 1, -60, 18, 48, -23, 38, 20, -70, 6, 42, -40, 44, 12, -138, 24, 48, -5, 57, 31, -90, 14, 54, -77, 72, 8, -100, 30, 60, -30, 62, 32, -184

Offset: 1

Michael Somos, Jul 30 2007

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

			G.f. = q + q^2 - 5*q^3 + q^4 + 6*q^5 - 5*q^6 + 8*q^7 + q^8 - 23*q^9 + 6*q^10 +...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.65).

Cf. A004016, A005882, A005928, A131943.

a[ n_] := If[ n < 1, 0, Sum[ d {0, 1, 0, -2, 0, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Nov 11 2015 *)
QP = QPochhammer; s = (1 - QP[q]^3*(QP[q^2]^3/(QP[q^3]*QP[q^6])))/(3 q) + O[q]^70; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, d*((d%6==1) + (d%6==5) - 2*(d%6==3))))};

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

Expansion of (1 - eta(q)^3 * eta(q^2)^3 / (eta(q^3) * eta(q^6))) / 3 in powers of q.
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 4- 3^(e+1), a(p^e) = (p^(e+1) - 1) / (p-1) if p>3.
G.f.: (1 - Product_{k>0} ((1 - x^k) * (1 -x^(2*k)))^3 / ((1 - x^(3*k)) * (1 - x^(6*k)))) / 3.
G.f.: Sum_{k>0} (6*k-1) * x^(6*k-1) / (1 - x^(6*k-1)) - 2 * (6*k-5) * x^(6*k-3) / (1 - x^(6*k-3)) + (6*k-5) * x^(6*k-5) / (1 - x^(6*k-5)).
-3 * a(n) = A131943(n) unless n=0.
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = 4*Pi^2/81 = 0.487387... . - Amiram Eldar, Jan 22 2024

A214456 Expansion of b(q^2) * (b(q) + 2 * b(q^4)) / 3 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -1, -3, 5, -3, -6, 15, -8, -3, 23, -18, -12, 15, -14, -24, 30, -3, -18, 69, -20, -18, 40, -36, -24, 15, -31, -42, 77, -24, -30, 90, -32, -3, 60, -54, -48, 69, -38, -60, 70, -18, -42, 120, -44, -36, 138, -72, -48, 15, -57, -93, 90, -42, -54, 231, -72, -24, 100

Offset: 0

Michael Somos, Jul 18 2012

sign

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

			1 - q - 3*q^2 + 5*q^3 - 3*q^4 - 6*q^5 + 15*q^6 - 8*q^7 - 3*q^8 + 23*q^9 + ...

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. A118271, A131943, A134077, A214361.

a[ n_] := SeriesCoefficient[ (9 EllipticTheta[ 3, 0, q^3]^4 - EllipticTheta[ 3, 0, q]^4) / 8, {q, 0, n}]
a[ n_] := SeriesCoefficient[ QPochhammer[q^2]^4 QPochhammer[-q^3, q^6]^3 /QPochhammer[-q, q^2], {q, 0, n}]
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] QPochhammer[ -q^3, q^6])^3 EllipticTheta[ 2, 0, (-q)^(1/2)]^4 / (16 (-q)^(1/2)), {q, 0, n}]

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

{a(n) = if( n<1, n==0, -sigma(n) + if( n%3==0, 9 * sigma(n/3)) + if( n%4==0, 4 * sigma(n/4)) + if( n%12==0, -36 * sigma(n/12)))}

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

Expansion of (9 * phi(q^3)^4 - phi(q)^4) / 8 = phi(q) * (psi(-q) * chi(q^3))^3 = psi(-q)^4 * (chi(q) * chi(q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^2)^2 * eta(q^4) * eta(q^6)^6 / ( eta(q^3) * eta(q^12))^3 in powers of q.
Euler transform of period 12 sequence [ -1, -3, 2, -4, -1, -6, -1, -4, 2, -3, -1, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 36 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A214361.
a(n) = -b(n) where b() is multiplicative with b(2^e) = 3 if e>0, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f.: Product_{k>0} (1 - x^(2*k))^4 * (1 + x^(6*k-3))^3 / (1 + x^(2*k-1)).
a(n) = (-1)^n * A118271(n). a(2*n) = a(4*n) = A131943(n). a(2*n + 1) = -A134077(n).

A226139 Expansion of b(-q) * b(q^2) in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, 3, -3, -15, -3, 18, 15, 24, -3, -69, -18, 36, 15, 42, -24, -90, -3, 54, 69, 60, -18, -120, -36, 72, 15, 93, -42, -231, -24, 90, 90, 96, -3, -180, -54, 144, 69, 114, -60, -210, -18, 126, 120, 132, -36, -414, -72, 144, 15, 171, -93, -270, -42, 162, 231, 216

Offset: 0

Michael Somos, May 27 2013

sign

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

Number 21 of the 126 eta-quotients listed in Table 1 of Williams 2012.

			1 + 3*q - 3*q^2 - 15*q^3 - 3*q^4 + 18*q^5 + 15*q^6 + 24*q^7 - 3*q^8 + ...

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
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A131943, A134077, A226132.

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

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

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

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

Expansion of = b(q^2) * (2 * b(q^4) - b(q)) = b(q^2)^4 / (b(q) * b(q^4)) in powers of q where b() is a cubic AGM function.
Expansion of (a(q) + a(q^2)) * (a(q^2) - 2 * a(q^4)) / 2 in powers of q where a() is a cubic AGM theta function.
Expansion of (psi(q)^3 / psi(q^3)) * (phi(-q^2)^3 / phi(-q^6)) in power of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2)^12 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)^4) in powers of q.
a(n) = 3 * b(n) where b(n) is multiplicative and b(2^e) = -1, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226132.
Euler transform of period 12 sequence [ 3, -9, 2, -6, 3, -6, 3, -6, 2, -9, 3, -4, ...].
G.f.: 1 + 3 * (Sum_{k>0} x^k / (1 - (-x)^k)^2 - 9 * x^(3*k) / (1 - (-x)^(3*k))^2).
G.f.: 1 + 3 * Sum_{k>0 not 2|k} k * (x^k * x^k / (1 + x^(2*k)) - 9 * x^(3*k) / (1 + x^(3*k))).
G.f.: Product_{k>0} (1 - x^(2*k))^12 * (1 + x^(6*k)) / ( (1 - x^k)^3 * (1 + x^(3*k))^3 * (1 - x^(3*k))^2 * (1 - x^(4*k))^3 ).
a(n) = (-1)^n * A131943(n). a(2*n) = A131943(n). a(2*n + 1) = 3 * A134077(n).

cp's OEIS Frontend

A186100 Expansion of 2 * a(q^2)^2 - a(q)^2 in powers of q where a() is a cubic AGM theta function.

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A132000 Expansion of (1/3) * b(q) * b(q^2) * c(q)^2 / c(q^2) in powers of q where b(), c() are cubic AGM functions.

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A131944 Expansion of (1 - b(q)*b(q^2)) / 3 where b() is a cubic AGM function. Expansion of (1 - eta(q)^3 * eta(q^2)^3 / (eta(q^3) * eta(q^6))) / 3 in powers of q.

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A214456 Expansion of b(q^2) * (b(q) + 2 * b(q^4)) / 3 in powers of q where b() is a cubic AGM theta function.

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A226139 Expansion of b(-q) * b(q^2) in powers of q where b() is a cubic AGM theta function.

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A131944 Expansion of (1 - b(q)b(q^2)) / 3 where b() is a cubic AGM function. Expansion of (1 - eta(q)^3 eta(q^2)^3 / (eta(q^3) * eta(q^6))) / 3 in powers of q.