A004007 -id:A004007 - cp's OEIS frontend

A109143 G.f.: cube root of theta series of E_6 lattice (cf. A004007).

1, 24, -486, 18960, -866028, 43409520, -2305522278, 127386486480, -7243559214894, 420974335099176, -24888628571711040, 1491922400816664432, -90454843306100805420, 5536766153219810009520, -341663245004722577661324, 21230836457057377337055600, -1327296238831338778081286796

Offset: 0

N. J. A. Sloane and Nadia Heninger, Aug 18 2005

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Vaclav Kotesovec, Table of n, a(n) for n = 0..540
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

terms = 17; QP = QPochhammer; s = (QP[q]^9/QP[q^3]^3 + 81 q QP[q^3]^9 / QP[q]^3)^(1/3) + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 08 2017, after Michael Somos *)

a(n) ~ -(-1)^n * c * d^n / n^(4/3), where d = 68.1926041339304593440433078460708736926328270041397481947611089811133711468... and c = 0.241679782171000058724170446760823529819630243406508395073992160578... - Vaclav Kotesovec, Dec 11 2017

A005129 Theta series of {E_6}* lattice.

Original entry on oeis.org

1, 0, 54, 72, 0, 432, 270, 0, 918, 720, 0, 2160, 936, 0, 2700, 2160, 0, 5184, 2214, 0, 5616, 3600, 0, 9504, 4590, 0, 9180, 6552, 0, 15120, 5184, 0, 14742, 10800, 0, 21600, 9360, 0, 19548, 12240, 0, 30240, 13500, 0, 28080, 17712, 0, 39744, 14760, 0, 32454

Offset: 0

N. J. A. Sloane

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

			G.f. = 1 + 54*x^2 + 72*x^3 + 432*x^5 + 270*x^6 + 918*x^8 + 720*x^9 + 2160*x^11 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 127.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
G. Nebe and N. J. A. Sloane, Home page for this lattice

Cf. A004007 (E_6).

A := Basis( ModularForms( Gamma1(3), 3), 51); A[1]; /* Michael Somos, Dec 28 2015 */

a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[(QPochhammer[x+A]^3 / QPochhammer[x^3+A])^3 + 9*x*(QPochhammer[x^3+A]^3 / QPochhammer[x+A])^3, {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 05 2015, adapted from 1st PARI script *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 / QPochhammer[ q^3])^3 + 9 q (QPochhammer[ q^3]^3 /QPochhammer[ q])^3, {q, 0, n}]; Table[a[n], {n, 0, 80}] (* Michael Somos, Dec 28 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 / eta(x^3 + A))^3 + 9 * x * (eta(x^3 + A)^3 / eta(x + A))^3, n))}; /* Michael Somos, Feb 28 2012 */

{a(n) = my(A, a1, p3); if( n<0, 0, A = x * O(x^n); a1 = sum( k=1, n, 6 * sumdiv(k, d, kronecker( d, 3)) * x^k, 1 + A); p3 = sum( k=1, n\3, -24 * sigma(k) * x^(3*k), 1 + A); polcoeff( (a1^3 + a1 * p3 - 4 * x * a1') / 2, n))}; /* Michael Somos, Feb 28 2012 */

Expansion of b(q)^3 + c(q)^3 / 3 in power of q where b(), c() are cubic AGM theta functions.

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

A181976 Expansion of a(q) * b(q)^2 in powers of q where a(), b() are cubic AGM theta functions.

Original entry on oeis.org

1, 0, -27, 72, 0, -216, 270, 0, -459, 720, 0, -1080, 936, 0, -1350, 2160, 0, -2592, 2214, 0, -2808, 3600, 0, -4752, 4590, 0, -4590, 6552, 0, -7560, 5184, 0, -7371, 10800, 0, -10800, 9360, 0, -9774, 12240, 0, -15120, 13500, 0, -14040, 17712, 0, -19872, 14760

Offset: 0

Michael Somos, Apr 04 2012

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

			G.f. = 1 - 27*q^2 + 72*q^3 - 216*q^5 + 270*q^6 - 459*q^8 + 720*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A004007.

eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q]^9 + 9*q*eta[q]^6*eta[q^9]^3)/eta[q^3]^3, {q, 0, 50}], q] (* G. C. Greubel, Aug 11 2018 *)

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

Expansion of b(q^3)^3 - 3 * b(q) * c(q^3)^2 in powers of q where b(), c() are cubic AGM theta functions.
Expansion of b(q^3)^2 * (b(q) + c(q^3)) in powers of q^3 where b(), c() are cubic AGM theta functions.
Expansion of (eta(q)^9 + 9 * q * eta(q)^6 * eta(q^9)^3) / eta(q^3)^3 in powers of q.
a(3*n + 1) = 0. a(3*n) = A004007(n).

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A109143 G.f.: cube root of theta series of E_6 lattice (cf. A004007).

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A005129 Theta series of {E_6}* lattice.

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A181976 Expansion of a(q) * b(q)^2 in powers of q where a(), b() are cubic AGM theta functions.

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