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A002607
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Glaisher's chi_8(n).
(Formerly M4994 N2150)
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1, 16, 0, 256, -1054, 0, 0, 4096, 6561, -16864, 0, 0, -478, 0, 0, 65536, -63358, 104976, 0, -269824, 0, 0, 0, 0, 720291, -7648, 0, 0, -1407838, 0, 0, 1048576, 0, -1013728, 0, 1679616, 925922, 0, 0, -4317184, 3577922, 0, 0, 0, -6915294, 0, 0, 0, 5764801, 11524656, 0, -122368
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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REFERENCES
| J. W. L. Glaisher, On the representation of a number as sum of 18 squares, Quart. J. Pure and Appl. Math. 38 (1907), 289-351 (see p. 304).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
W. Stein, Modular Forms Database.
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FORMULA
| Expansion of newform of degree 1, level 4, weight 9 and nontrivial character in powers of q. - Michael Somos Mar 09 2006
Expansion of Jacobi ((2kk')^2+(kk')^4)(2K/pi)^9/64 in powers of q. - Michael Somos Mar 09 2006
Expansion of F(phi(q)^4,q*psi(q^2)^4) in powers of q where F(u,v)=sqrt(u)*v*(u-16*v)*(u^2+4*u*v-64*v^2) and phi(),psi() are Ramanujan theta functions. - Michael Somos Mar 09 2006
a(n) is multiplicative with a(2^e) = 16^e, a(p^e) = p^(4e)*(1+(-1)^e)/2 if p == 3 (mod 4), a(p^e) = a(p)*a(p^(e-1)) - p^8*a(p^(e-2)) if p == 1 (mod 4) . - Michael Somos Mar 09 2006
G.f.: (t''''*t -28*t'''*t' +35*t''^2)/2 where t=phi(q) and f' := q*df/dq . - Michael Somos Mar 09 2006
G.f.: ( Sum_{j,k} (j+i*k)^8* x^(j^2+k^2) )/4 . a(4n+3)=0.
Expansion of q* f(-q^2)^18* (chi(q)^12 +4*q/ chi(q)^12) in powers of q where f(), chi() are Ramanujan theta functions. - Michael Somos Jul 25 2007
G.f. is Fourier series of a weight 9 level 4 cusp form. f(-1/ (4 t)) = i (-2 t)^9 f(t) where q = exp(2 pi i t). - Michael Somos Jul 25 2007
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EXAMPLE
| q + 16*q^2 + 256*q^4 - 1054*q^5 + 4096*q^8 + 6561*q^9 - 16864*q^10 - ...
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PROG
| (PARI) {a(n)=local(m); if(n<1, 0, m=sqrtint(n); polcoeff( sum(j=-m, m, sum(k=-m, m, (j+I*k)^8* x^(j^2+k^2), x*O(x^n)))/4, n))} /* Michael Somos Mar 09 2006 */
(PARI) {a(n)= local(A, B); if(n<1, 0, n--; A= x*O(x^n); B= (eta(x^2+A)^2/ eta(x+A)/ eta(x^4+A))^12; polcoeff( eta(x^2+A)^18* (B +4*x/B), n))} /* Michael Somos Jul 25 2007 */
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CROSSREFS
| Sequence in context: A188784 A123935 A169764 * A198804 A111979 A173436
Adjacent sequences: A002604 A002605 A002606 * A002608 A002609 A002610
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KEYWORD
| sign,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by Michael Somos, Mar 09, 2006
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