A004010 -id:A004010 - cp's OEIS frontend

A107658 Coefficients of 6th root of theta series of Coxeter-Todd lattice K_{12} (see A004010).

1, 0, 126, 672, -36288, -413280, 15087870, 275661792, -6846707322, -186737716704, 3093536396160, 126405712075104, -1274633447433024, -84873293805379968, 385697576191762044, 56246329449791661600, 31646424393253329408

Offset: 0

N. J. A. Sloane and Michael Somos, Jun 07 2005

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..693
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.

Cf. A004010.

terms = 17; QP = QPochhammer; s = ((QP[q]^18*(4 - 4*(1 + (9*q*QP[q^9]^3) / QP[q]^3)^3 + 3*(1 + (9*q*QP[q^9]^3)/QP[q]^3)^6))/(3*QP[q^3]^6))^(1/6) + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 08 2017, after Michael Somos *)

A320686 Inverse Euler transform of A004010.

Original entry on oeis.org

0, 756, 4032, -265734, -2987712, 120604680, 2176735680, -58263976134, -1563340453248, 27722120100948, 1105815958027200, -12029301541618956, -769283790627284352, 3952625120472002580, 525306588856752370752, 41570815360527775098, -351118365555207656907648

Offset: 1

Seiichi Manyama, Oct 19 2018

sign

a(n) is a multiple of 6.

			1 + 756*q^2 + 4032*q^3 + 20412*q^4 + ... = (1-q^2)^(-756) * (1-q^3)^(-4032) * (1-q^4)^265734 * ... .

Seiichi Manyama, Table of n, a(n) for n = 1..695
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Coxeter-Todd Lattice

Cf. A004010, A107658.

A004046 Theta series of extremal 3-modular even 24-dimensional lattice with minimal norm 6 and det = 3^12.

Original entry on oeis.org

1, 0, 0, 26208, 530712, 6368544, 47331648, 256864608, 1116087336, 4092877152, 12996075456, 37058557536, 96952754808, 232778774592, 526258264896, 1128148021728, 2286143305992, 4451523096384, 8386247967552, 15130902687264, 26614339616592, 45684687301344

Offset: 0

N. J. A. Sloane

nonn

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 729 (t/i)^12 f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 21 2015

			G.f. = 1 + 26208*x^3 + 530712*x^4 + 6368544*x^5 + 47331648*x^6 + ...
G.f. = 1 + 26208*q^6 + 530712*q^8 + 6368544*q^10 + 47331648*q^12 + ...

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
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 lattice
H.-G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory, 54 (1995), 190-202.
N. J. A. Sloane, Seven Staggering Sequences.

Cf. A107657.

A := Basis( ModularForms( Gamma1(3), 12), 22);  A[1] + 26208*A[4] + 530712*A[5]; /* Michael Somos, Dec 21 2015 */

a[ n_] := With[ {U1 = QPochhammer[ q]^3, U3 = QPochhammer[ q^3]^3, U9 = QPochhammer[ q^9]^3}, With[ {z = ( 1 + 9 q U9/U1)^3}, SeriesCoefficient[ (U1^3/U3)^4 (27 z^4 - 72 z^3 + 64 z^2 + 16 z - 8) / 27, {q, 0, n}]]]; (* Michael Somos, Dec 25 2015 *)

th3 = sum(n=1,noo\2, 2*x^(4*n^2), 1+A);
th4 = sum(n=1,noo\2, (-1)^n*2*x^(4*n^2), 1+A);
th2 = sum(n=0,noo\2, 2*x^(4*n^2+4*n+1), A);
chk("th3^4 == th4^4+th2^4");
/* A004016(x^4) */
phi0 = th2*subst(th2,x,x^3)+ th3*subst(th3,x,x^3);
/* 2*x*A033762(x^2) */
phi1 = th2*subst(th3,x,x^3)+ th3*subst(th2,x,x^3);
/* A004010(x^2) */
K_12 = phi0^6+45*phi0^2*phi1^4+18*phi1^6;
a=phi0;b=phi1;
A004046=a^12-9/2*a^8*b^4+414*a^6*b^6+1458*a^4*b^8+1998*a^2*b^10+459/2*b^12;

{a(n) = my(A, U1, U3, U9, z); if( n<0, 0, A = x * O(x^n); U1 = eta(x + A)^3; U3 = eta(x^3 + A)^3; U9 = eta(x^9 + A)^3; z = (1 + 9 * x * U9/U1)^3; polcoeff( (U1^3/U3)^4 * (27*z^4 - 72*z^3 + 64*z^2 + 16*z - 8) / 27, n))}; /* Michael Somos, Dec 25 2015 */

Theta series = a^12 - 9/2*a^8*b^4 + 414*a^6*b^6 + 1458*a^4*b^8 + 1998*a^2*b^10 + 459/2*b^12 (see PARI code for details).

G.f.: (27*a(x)^12 - 72*a(x)^9*b(x)^3 + 64*a(x)^6*b(x)^6 + 16*a(x)^3*b(x)^9 - 8*b(x)^12) / 27 where a(), b() are cubic AGM theta functions, - Michael Somos, Dec 25 2015

PARI code from Michael Somos, Jun 07 2005

A015235 Theta series of lattice Kappa_8.

Original entry on oeis.org

1, 0, 132, 192, 828, 1152, 2796, 2880, 6828, 5376, 14904, 10944, 20772, 18432, 40224, 25920, 53964, 41472, 76452, 58176, 107784, 69504, 156816, 101376, 163284, 131328, 259032, 147072, 295200, 206208, 357480, 250560, 432780, 269568, 576072, 365184, 555804, 426240

Offset: 0

N. J. A. Sloane

nonn

			G.f. = 1 + 132*q^4 + 192*q^6 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 161.

Andy Huchala, Table of n, a(n) for n = 0..20000
G. Nebe and N. J. A. Sloane, Home page for this lattice

Cf. A015236 (K_7), A015233 (K_9), A015232 (K_10), A015229 (K_11), A004010 (K_12), A029897 (K_13), A047628 (K_14).

L = [1, 0, 132, 192, 828, 1152, 2796, 2880, 6828, 5376]
M = ModularForms(Gamma0(12),4)
bases = [.q_expansion(35) for  in M.integral_basis()]
f = sum(x*y for (x,y) in zip(bases,L)); list(f) # Andy Huchala, Jul 23 2021

More terms from Sean A. Irvine, Feb 26 2020

A320676 Expansion of (r(q) * s(q))^3 where r(), s() are cubic AGM theta functions.

Original entry on oeis.org

1, 9, -27, -261, 765, 2214, -11529, 11304, 24813, -81423, 71118, 106812, -354609, 262350, 385992, -1049166, 739917, 990306, -2713203, 1709604, 2287710, -5646600, 3707532, 4448952, -11344833, 6737319, 8450838, -19943757, 12298248, 14238558, -34639974, 19856736

Offset: 0

Seiichi Manyama, Oct 19 2018

sign

Cubic AGM theta functions: r(q) (see A004016), s(q) (A005928), t(q) (A005882).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A004010, A008654, A281722.

Expansion of (eta(q)^3 * (eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3)^2)^3 in powers of q.

A320677 Expansion of s(q)^6 where s() is cubic AGM theta functions.

Original entry on oeis.org

1, -18, 135, -504, 657, 2052, -10071, 12384, 20277, -83610, 72090, 122040, -355581, 245124, 379512, -1050624, 770589, 966492, -2700081, 1724616, 2287062, -5636880, 3616164, 4471632, -11385657, 6820722, 8554194, -19963440, 12302568, 14113332, -34631226, 19737936

Offset: 0

Seiichi Manyama, Oct 19 2018

sign

Cubic AGM theta functions: r(q) (see A004016), s(q) (A005928), t(q) (A005882).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

s(q)^m: A005928 (m=1), A242874 (m=2), A109041 (m=3), A133078 (m=4), this sequence (m=6).

Cf. A004010, A109041.

Expansion of (eta(q)^3/eta(q^3))^6 in powers of q.

cp's OEIS Frontend

A107658 Coefficients of 6th root of theta series of Coxeter-Todd lattice K_{12} (see A004010).

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A320686 Inverse Euler transform of A004010.

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A004046 Theta series of extremal 3-modular even 24-dimensional lattice with minimal norm 6 and det = 3^12.

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A015235 Theta series of lattice Kappa_8.

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A320676 Expansion of (r(q) * s(q))^3 where r(), s() are cubic AGM theta functions.

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A320677 Expansion of s(q)^6 where s() is cubic AGM theta functions.

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