A106402 -id:A106402 - cp's OEIS frontend

A109041 Expansion of eta(q)^9 / eta(q^3)^3 in powers of q.

1, -9, 27, -9, -117, 216, 27, -450, 459, -9, -648, 1080, -117, -1530, 1350, 216, -1845, 2592, 27, -3258, 2808, -450, -3240, 4752, 459, -5409, 4590, -9, -5850, 7560, -648, -8658, 7371, 1080, -7776, 10800, -117, -12330, 9774, -1530, -11016, 15120, 1350, -16650

Offset: 0

Michael Somos, Jun 17 2005

sign

Number 4 of the 74 eta-quotients listed in Table I of Martin (1996).

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

			G.f. = 1 - 9*q + 27*q^2 - 9*q^3 - 117*q^4 + 216*q^5 + 27*q^6 - 450*q^7 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 313, Equ. (14.2.13).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
George E. Andrews and Bruce C. Berndt, Your Hit Parade: The Top Ten Most Fascinating Formulas in Ramanujan's Lost Notebook, Notices Amer. Math. Soc., 55 (No. 1, 2008), 18-30. See p. 23, Equation (27).
Jonathan M. Borwein and Peter B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. See p. 697.
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
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

Cf. A005928, A008654, A103440, A106402, A134340.

A := Basis( ModularForms( Gamma1(3), 3), 44); A[1] - 9*A[2]; /* Michael Somos, May 18 2015 */

a[ n_] := If[ n < 1, Boole[ n == 0], - 9 DivisorSum[ n, #^2 KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Jul 19 2012 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 / QPochhammer[ q^3])^3, {q, 0, n}]; (* Michael Somos, Jul 19 2012 *)

{a(n) = if( n<1, n==0, -9 * sumdiv( n, d, d^2 * kronecker(-3, d)))};

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

G.f.: Product_{k>0} (1 - x^k)^9 / (1 - x^3)^3 = 1 - 9 * Sum_{k>0} x^k * (1 - x^k -6 * x^(2*k) - x^(3*k) + x^(4*k)) / (1 + x^k + x^(2*k))^3.
Expansion of b(q)^3 in powers of q where b() is a cubic AGM theta function.
Euler transform of period 3 sequence [ -9, -9, -6, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + u*w * (u + 6*v - 8*w).
Given A = A0 + A1 + A2 is the 3-section, then 0 = A1^3 + A2^3 - 3*A0*A1*A2. A0 = A(q^3) = b(q^3)^3, A1 = -3 * a(q^3)^2 * c(q^3), A2 = 3 * a(q^3) * c(q^3)^2 where a(), b(), c() are cubic AGM theta functions.
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 19683^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A106402. - Michael Somos, Mar 11 2012
a(n) = -9 * A103440(n) unless n = 0. a(6*n + 5) = 216 * A134340(n).
A008654(n) = a(n) + 27 * A106402(n) is the identity a(q)^3 = b(q)^3 + c(q)^3. - Michael Somos, Jul 19 2012
a(n) = -9 * b(n) where b(n) is multiplicative with a(0) = 1, b(p^e) = 1, if p=3, b(p^e) = b(p) * b(p^(e-1)) + Kronecker(-3, p) * p^2 * b(p^(e-2)) otherwise. - Michael Somos, May 18 2015
Convolution cube of A005928. - Michael Somos, May 18 2015

A198956 q-expansion of modular form psi_0^4/t_{3B}.

Original entry on oeis.org

0, 1, 9, 27, 73, 126, 243, 344, 585, 729, 1134, 1332, 1971, 2198, 3096, 3402, 4681, 4914, 6561, 6860, 9198, 9288, 11988, 12168, 15795, 15751, 19782, 19683, 25112, 24390, 30618, 29792, 37449, 35964, 44226, 43344, 53217, 50654, 61740, 59346, 73710, 68922, 83592

Offset: 0

N. J. A. Sloane, Nov 01 2011

nonn

psi_0 is given in A004016, t_{3B} in A198955.

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

			G.f. = q + 9*q^2 + 27*q^3 + 73*q^4 + 126*q^5 + 243*q^6 + 344*q^7 + 585*q^8 + 729*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]

Cf. A001158, A004016, A106402, A198955, A215711.

Basis( ModularForms( Gamma0(3), 4), 43)[2]; /* Michael Somos, Dec 27 2014 */

a[ n_] := SeriesCoefficient[ q QPochhammer[ q^3]^8 (1 + 9 q (QPochhammer[ q^9] / QPochhammer[ q])^3), {q, 0, n}]; (* Michael Somos, Dec 27 2014 *)

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^8 * (1 + 9 * x * (eta(x^9 + A) / eta(x + A))^3), n))}; /* Michael Somos, Aug 23 2012 */

ModularForms( Gamma0(3), 4, prec=43).1;# Michael Somos, May 23 2014

Expansion of a(q) * (c(q) / 3)^3 in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, Aug 23 2012
Expansion of eta(q^3)^8 * (1 + 9 * (eta(q^9) / eta(q))^3) in powers of q. - Michael Somos, Aug 23 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = (1/3) (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A215711. - Michael Somos, Aug 23 2012
Convolution of A004016 and A106402. - Michael Somos, Aug 23 2012
Conjecture: Multiplicative with a(3^e) = 3^e, a(p^e) = sigma_3(p^e) for prime p <> 3. - Andrew Howroyd, Aug 08 2018

A198958 q-expansion of modular form psi_0^6/t_{3B}^2.

Original entry on oeis.org

0, 0, 1, 6, 27, 80, 207, 432, 863, 1512, 2646, 4144, 6585, 9504, 14216, 19476, 27783, 36384, 49977, 63504, 84722, 104736, 136188, 165056, 210717, 250560, 314270, 367902, 455544, 525808, 642762, 733968, 888087, 1003608, 1201554, 1347232

Offset: 0

N. J. A. Sloane, Nov 01 2011

nonn

psi_0 is given in A004016, t_{3B} in A198955.

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

			G.f. = q^2 + 6*q^3 + 27*q^4 + 80*q^5 + 207*q^6 + 432*q^7 + 863*q^8 + 1512*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]

Cf. A106402.

A := Basis( ModularForms( Gamma1(3), 6), 36); A[3]; /* Michael Somos, Feb 22 2015 */

a[ n_] := SeriesCoefficient[ q^2 (QPochhammer[ q^3]^3 / QPochhammer[ q])^6, {q, 0, n}]; (* Michael Somos, Feb 22 2015 *)

{a(n) = my(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( (eta(x^3 + A)^3 / eta(x + A))^6, n))}; /* Michael Somos, Jun 07 2012 */

Expansion of (c(q) / 3)^6 in powers of q where c() is a cubic AGM theta function. - Michael Somos, Jun 07 2012
Expansion of (eta(q^3)^3 / eta(q))^6 in powers of q.
G.f.: (Product_{k>0} (1 - x^(3*k))^3 / (1 - x^k))^6. - Michael Somos, Jun 07 2012
Convolution square of A106402. - Michael Somos, Dec 26 2015

A181977 Expansion of b(q) * c(q^3)^2 / 9 in powers of q where b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, -3, 0, 8, -9, 0, 17, -27, 0, 40, -39, 0, 50, -72, 0, 96, -81, 0, 104, -150, 0, 176, -153, 0, 170, -243, 0, 280, -216, 0, 273, -360, 0, 400, -351, 0, 362, -510, 0, 560, -450, 0, 520, -648, 0, 736, -615, 0, 601, -864, 0, 936, -729, 0, 850, -1086, 0, 1160

Offset: 2

Michael Somos, Apr 04 2012

sign

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

			G.f. = q^2 - 3*q^3 + 8*q^5 - 9*q^6 + 17*q^8 - 27*q^9 + 40*q^11 - 39*q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 2..2500

Cf. A106402.

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]* eta[q^9]^2/eta[q^3])^3, {q, 0, 50}], q] (* G. C. Greubel, Aug 11 2018 *)

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

Expansion of (eta(q) * eta(q^9)^2 / eta(q^3))^3 in powers of q.
Euler transform of period 9 sequence [-3, -3, 0, -3, -3, 0, -3, -3, -6, ...].
a(3*n + 1) = 0. a(3*n) = -3 * A106402(n).

A266288 Expansion of a(q)^2 * (c(q)/3)^3 in powers of q where a(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, 15, 81, 241, 624, 1215, 2402, 3855, 6561, 9360, 14640, 19521, 28562, 36030, 50544, 61681, 83520, 98415, 130322, 150384, 194562, 219600, 279840, 312255, 390001, 428430, 531441, 578882, 707280, 758160, 923522, 986895, 1185840, 1252800, 1498848, 1581201

Offset: 1

Michael Somos, Dec 26 2015

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

Convolution of A008653 and A106402.

			G.f. = x + 15*x^2 + 81*x^3 + 241*x^4 + 624*x^5 + 1215*x^6 + 2402*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A004016, A005882, A005928, A008653, A106402, A344778.

A := Basis( ModularForms( Gamma1(3), 5),37);  A[2];

a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (With[{s = {1, -1, 0}[[Mod[#, 3, 1]]]}, ((#^4)^(#2 + 1) - s^(#2 + 1)) / (#^4 - s)] & @@@ FactorInteger[n])];

{a(n) = my(A, U1, u3, U9); if( n<1, 0, n--; A = x * O(x^n); U1 = eta(x + A)^3; u3 = eta(x^3 + A); U9 = eta(x^9 + A)^3; polcoeff( U1 * u3^7 * (1 + 9*x*U9/U1)^2, n))};

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

a(n) is multiplicative with a(p^e) = ((p^4)^(e+1) - s^(e+1)) / (p^4 - s) where s = 0 if p = 3, s = 1 if p == 1 (mod 3), s = -1 if p == 2 (mod 3).

Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 4*Pi^5/(729*sqrt(3)) = 0.9694405... (A344778). - Amiram Eldar, Nov 09 2023

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A109041 Expansion of eta(q)^9 / eta(q^3)^3 in powers of q.

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A198956 q-expansion of modular form psi_0^4/t_{3B}.

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A198958 q-expansion of modular form psi_0^6/t_{3B}^2.

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A181977 Expansion of b(q) * c(q^3)^2 / 9 in powers of q where b(), c() are cubic AGM theta functions.

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A266288 Expansion of a(q)^2 * (c(q)/3)^3 in powers of q where a(), c() are cubic AGM theta functions.

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