|
Search: id:A030212
|
|
|
| A030212 |
|
Expansion of eta(q)^4*eta(q^2)^2*eta(q^4)^4. |
|
+0
3
|
|
| 0, 1, -4, 0, 16, -14, 0, 0, -64, 81, 56, 0, 0, -238, 0, 0, 256, 322, -324, 0, -224, 0, 0, 0, 0, -429, 952, 0, 0, 82, 0, 0, -1024, 0, -1288, 0, 1296, 2162, 0, 0, 896, -3038, 0, 0, 0, -1134, 0, 0, 0, 2401, 1716, 0, -3808, 2482, 0, 0, 0, 0, -328, 0, 0, -6958, 0, 0, 4096, 3332, 0, 0, 5152, 0, 0
(list; graph; listen)
|
|
|
OFFSET
|
0,3
|
|
|
COMMENT
|
Euler transform of period 4 sequence [ -4,-6,-4,-10,...]. - Michael Somos, Jul 17 2004
Multiplicative with a(2^e)=(-4)^e, a(p^e)=p^2e if e even else 0 for p=3 mod 4.
Called chi_4(n) by Glaisher and Hardy.
|
|
REFERENCES
|
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135 section 9.3. MR0106147 (21 #4881)
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 175, 4.7 Exercise 5. MR1471703 (98g:14032)
J. W. L. Glaisher, On the representation of a number as sum of 2,4,6,8... squares, Quart. J. Math. 38 (1907), 1-62 (see p. 34).
|
|
FORMULA
|
G.f. x(Product_{k>0} (1-x^k)^4(1-x^(2k))^2(1-x^(4k))^4). a(4k+3)=0. a(4k)=16a(k).
G.f.: (t*t''-3(t')^2)/2 where t=theta_3(x) (A000122) and t' := x*(dt/dx), t'' := (t')'. - Michael Somos Nov 08 2005
Given A=A0+A1+A2+A3 is the 4-section, then 0=(A0^2-A2^2)^2+4*A0*A2*A1^2 . - Michael Somos Mar 08 2006
|
|
EXAMPLE
|
q -4*q^2 +16*q^4 -14*q^5 -64*q^8 +81*q^9 +56*q^10 -238*q^13 +...
|
|
PROGRAM
|
(PARI) a(n)=local(A); if(n<1, 0, n--; A=x^n*O(x); polcoeff((eta(x+A)*eta(x^4+A))^4*eta(x^2+A)^2, n)) /* Michael Somos, Jul 17 2004 */
(PARI) {a(n)=local(r); if(n<1, 0, r=sqrtint(n); sum(x=-r, r, sum(y=-r, r, if(x^2+y^2==n, (x+I*y)^4) ))/4 )} /* Michael Somos Sep 12 2005 */
|
|
CROSSREFS
|
Sequence in context: A134746 A003195 A086262 this_sequence A007216 A057378 A002979
Adjacent sequences: A030209 A030210 A030211 this_sequence A030213 A030214 A030215
|
|
KEYWORD
|
sign,mult
|
|
AUTHOR
|
njas
|
|
|
Search completed in 0.002 seconds
|
