|
Search: id:A002653
|
|
|
| A002653 |
|
Expansion of (theta_3(z)*theta_3(7z)+theta_2(z)*theta_2(7z))^3. |
|
+0
3
|
|
| 1, 6, 24, 56, 114, 168, 280, 294, 444, 390, 840, 636, 1176, 1176, 1512, 1008, 1782, 2016, 1896, 2520, 3528, 2408, 3216, 2796, 4760, 3174, 5880, 4592, 6258, 4380, 5040, 6720, 7200, 6832, 10080, 7224, 8082, 7164, 12600, 7056, 14280, 11760, 12040, 9756
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
Theta series of Kleinian lattice (Z[ (-1+sqrt(-7))/2 ])^3 in 3 complex (or 6 real) dimensions.
|
|
REFERENCES
|
H. H. Chan and Y. L. Ong, On Eisenstein series and Sum_{m,n} q^(m^2+mn+2n^2), Proc. Amer. Math. Soc. 127 (1999), no. 6, 1735-1744, See page 1737. MR1600120 (99i:11029)
N. Elkies, The Klein quartic in number theory, pp. 51-101 of S. Levy, ed., The Eightfold Way, Cambridge Univ. Press, 1999. MR1722413 (2001a:11103)
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 53.
|
|
LINKS
|
N. Elkies, The Klein quartic in number theory
|
|
FORMULA
|
G.f.: (theta_3(z)*theta_3(7z) + theta_2(z)*theta_2(7z))^3.
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w) is a homogenous degree 6 polynomial with 28 terms. - Michael Somos Jun 03 2005
Expansion of (eta(q)^8 + 13 * eta(q)^4 * eta(q^7)^4 + 49 * eta(q^7)^8) / ( eta(q) * eta(q^7) ) in power of q. - Michael Somos Mar 11 2008
Expansion of (eta(q)^8 + 13 * (eta(q) * eta(q^7))^4 + 49 * eta(q^7)^8) / ( eta(q) * eta(q^7) ) in powers of q. - Michael Somos Mar 17 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 343^(1/2) (t/i)^3 f(t) where q = exp(2 pi i t).
|
|
EXAMPLE
|
1 + 6*q + 24*q^2 + 56*q^3 + 114*q^4 + 168*q^5 + 280*q^6 + 294*q^7 + ...
|
|
PROGRAM
|
(PARI) {a(n)=local(A, t2, t3); if(n<1, n==0, A=x*O(x^n); t2=sum(k=1, (sqrtint(4*n+1)+1)\2, 2*x^(k*k-k), A); t3=sum(k=1, sqrtint(n), 2*x^(k*k), 1+A); A=x*O(x^(n\7)); polcoeff( (t3*subst(t3+A, x, x^7)+x^2*t2*subst(t2+A, x, x^7))^3, n))} /* Michael Somos Jun 03 2005 */
(PARI) {a(n) = local(A, t1, t7); if( n<0, 0, A = x * O(x^n); t1 = eta(x + A)^4; t7 = eta(x^7 + A)^4; polcoeff( (t1^2 + 13 * x * t1 * t7 + 49 * x^2 * t7^2) / (t1 * t7)^(1/4), n))} /* Michael Somos Mar 11 2008 */
(PARI) {a(n) = local(A, A1, A7); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A); A7 = eta(x^7 + A); polcoeff( (A1^8 + 13 * x * (A1 * A7)^4 + 49 * x^2 * A7^8) / (A1 * A7), n))} /* Michael Somos Mar 17 2008 */
|
|
CROSSREFS
|
Cf. A002652.
Adjacent sequences: A002650 A002651 A002652 this_sequence A002654 A002655 A002656
Sequence in context: A033581 A009943 A028595 this_sequence A086768 A007531 A130669
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
njas
|
|
|
Search completed in 0.002 seconds
|
