A121596 -id:A121596 - cp's OEIS frontend

A121590 Expansion of (eta(q^3) / eta(q))^12 in powers of q.

1, 12, 90, 508, 2391, 9828, 36428, 124188, 395199, 1186344, 3387252, 9257364, 24343037, 61848096, 152356032, 364959196, 852243948, 1944226476, 4341094220, 9502198728, 20419293123, 43131708720, 89656112256, 183580652340

Offset: 1

Michael Somos, Aug 09 2006

nonn

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

			G.f. = q + 12*q^2 + 90*q^3 + 508*q^4 + 2391*q^5 + 9828*q^6 + 36428*q^7 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

Cf. A000726, A030182, A128758.

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3] / QPochhammer[ q])^12, {q, 0, n}]; (* Michael Somos, Aug 09 2015 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(3*k)) / (1 - x^k))^12, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)

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

Expansion of (c(q) / (3 * b(q)))^3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 16 2012
Euler transform of period 3 sequence [ 12, 12, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^3 + v^3 - u*v - 24*u*v * (u + v) - 729*u^2*v^2.
G.f.: x * (Product_{k>0} (1 - x^(3*k)) / (1 - x^k))^12.
Convolution inverse of A030182. - Michael Somos, Jun 16 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^-6 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A030182.
Convolution 12th power of A000726, cube of A128758, square of A121596. - Michael Somos, Aug 09 2015
a(n) ~ exp(4*Pi*sqrt(n/3)) / (729 * sqrt(2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
a(1) = 1, a(n) = (12/(n-1))*Sum_{k=1..n-1} A046913(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017

A007260 McKay-Thompson series of class 6a for Monster.

Original entry on oeis.org

1, -33, -153, -713, -2550, -7479, -20314, -51951, -122229, -276656, -601068, -1254105, -2541531, -5018721, -9647991, -18168984, -33554784, -60818040, -108471674, -190607871, -330140403, -564580142, -953980392, -1593599832, -2634301308, -4311874755, -6991318008

Offset: 0

N. J. A. Sloane

sign

A more correct name would be: Expansion of replicable function of class 6a. See Alexander et al., 1992. - N. J. A. Sloane, Jun 12 2015

			G.f. = 1 - 33*x - 153*x^2 - 713*x^3 - 2550*x^4 - 7479*x^5 - 20314*x^6 + ...
T6a = 1/q - 33*q - 153*q^3 - 713*q^5 - 2550*q^7 - 7479*q^9 - 20314*q^11 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..499 from G. A. Edgar)
D. Alexander, C. Cummins, J. McKay and C. Simons, Completely replicable functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
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.]
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Index entries for McKay-Thompson series for Monster simple group

Cf. A007242, A007250, A007262, A121596, A258917.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^3])^6 - 27 x (QPochhammer[ x^3] / QPochhammer[ x])^6, {x, 0, n} ]; (* Michael Somos, Jun 14 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^3 + A))^6 - 27 * x * (eta(x^3 + A) / eta(x + A))^6, n))}; /* Michael Somos, Jun 14 2015 */

{ my(q='q+O('q^66), t=(eta(q)/eta(q^3))^6 ); Vec( t - 27*q/t ) } \\ Joerg Arndt, Apr 02 2017

Expansion of q * ((eta(q^2) / eta(q^6))^6 - 27 * (eta(q^6) / eta(q^2))^6) in powers of q^2. - Michael Somos, Jun 14 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = -1 / f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 14 2015
a(n) = A007262(n) - 27 * A121596(n-1). - Michael Somos, Jun 14 2015
Convolution square is A258917. - Michael Somos, Jun 14 2015
a(n) ~ -exp(2*Pi*sqrt(2*n/3)) / (2^(3/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017

A007261 McKay-Thompson series of class 6b for the Monster group.

Original entry on oeis.org

1, 21, 171, 745, 2418, 7587, 20510, 51351, 122715, 277384, 598812, 1255761, 2543973, 5011725, 9653013, 18176040, 33535032, 60831648, 108490390, 190557015, 330174837, 564626278, 953857104, 1593681480, 2634409140, 4311592119, 6991502688, 11237020682, 17909802270

Offset: 0

N. J. A. Sloane

nonn

From Gary W. Adamson, Jul 21 2009: (Start)
(1 + 21x + 171x^2 + 745x^3 + ...)^2 = (1 + 42x + 783x^2 + 8672x^3 + ...)
where A030197 = (1, 42, 783, 8672, 65367, ...). (End)

			1 + 21*x + 171*x^2 + 745*x^3 + 2418*x^4 + 7587*x^5 + 20510*x^6 + 51351*x^7 + ...
T6b = 1/q + 21*q + 171*q^3 + 745*q^5 + 2418*q^7 + 7587*q^9 + 20510*q^11 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
J. McKay and H. Strauss, The q-series of monstrous moonshine and the decomposition of the head characters, Comm. Algebra 18 (1990), no. 1, 253-278.
Index entries for McKay-Thompson series for Monster simple group

Cf. A030197. - Gary W. Adamson, Jul 21 2009

Cf. A058537.

a[0] = 1; a[n_] := Module[{A = x*O[x]^n}, A = (QPochhammer[x^3 + A] / QPochhammer[x + A])^12; SeriesCoefficient[Sqrt[(1 + 27*x*A)^2/A], n]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 06 2015, adapted from Michael Somos's PARI script *)
CoefficientList[Series[(QPochhammer[x, x]^3 + 9*x*QPochhammer[x^9, x^9]^3)^3 / (QPochhammer[x, x]^3*QPochhammer[x^3, x^3]^6), {x, 0, 50}], x] (* Vaclav Kotesovec, Nov 07 2015 *)
nmax = 30; CoefficientList[Series[Product[(1 - x^k)^6/(1 - x^(3*k))^6, {k, 1, nmax}] + 27*x*Product[(1 - x^(3*k))^6/(1 - x^k)^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2017 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^3 + A) / eta(x + A))^12; polcoeff( sqrt((1 + 27 * x * A)^2 / A), n))} /* Michael Somos, Jun 16 2012 */

N=66; q='q+O('q^N); t=(eta(q) / eta(q^3))^6; Vec(t + 27*q/t) \\ Joerg Arndt, Mar 11 2017

Expansion of (27 * x * (b(x)^3 + c(x)^3)^2 / (b(x) * c(x))^3)^(1/2) in powers of x where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 16 2012
Convolution cube of A058537. - Michael Somos, Aug 20 2012
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Nov 07 2015
Expansion of q^(1/2) * (eta(q)^6/eta(q^3)^6 + 27*eta(q^3)^6/eta(q)^6) in powers of q. - G. A. Edgar, Mar 10 2017
a(n) = A007262(n) + 27 * A121596(n-1). - Sean A. Irvine, Nov 26 2017

A296162 a(n) = [x^n] Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 1, 5, 19, 89, 406, 1913, 9073, 43505, 209971, 1019390, 4971781, 24343037, 119579006, 589050663, 2908727839, 14393759457, 71360342129, 354372852011, 1762416036422, 8776797353574, 43761058841638, 218431753457637, 1091385769314272, 5458068218285909, 27319061357620056

Offset: 0

Ilya Gutkovskiy, Dec 06 2017

nonn

Eric Weisstein's World of Mathematics, Partition Function b_k

Cf. A000726, A008485, A121596, A128758, A270913, A285927, A296044, A296163.

Table[SeriesCoefficient[Product[((1 - x^(3 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
(* Calculation of constant d: *) With[{k = 3}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} (1 + x^k + x^(2*k))^n.

a(n) ~ c * d^n / sqrt(n), where d = 5.1069752682291604222843644516987970799026747758649349... and c = 0.271879273312907861082536692355942116774864... - Vaclav Kotesovec, May 13 2018

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A121590 Expansion of (eta(q^3) / eta(q))^12 in powers of q.

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A007260 McKay-Thompson series of class 6a for Monster.

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A007261 McKay-Thompson series of class 6b for the Monster group.

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A296162 a(n) = [x^n] Product_{k>=1} ((1 - x^(3*k))/(1 - x^k))^n.

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