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A002607
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Glaisher's chi_8(n).
(Formerly M4994 N2150)
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4
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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
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OFFSET
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1,2
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COMMENTS
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REFERENCES
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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). [The whole 1907 volume of The Quarterly Journal of Pure and Applied Mathematics, volume 38, is freely available from Google Books]
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
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FORMULA
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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 ((2*k(q)*k'(q))^2 + (k(q)*k'(q))^4) * (K(q) / (pi/2))^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^(4*e) * (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) where a(p) = 2 * Re( (x + i*y)^8 ) and p = x^2 + y^2 with even x. - Michael Somos, Nov 18 2014
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. where i^2 = -1. a(2*n) = 16*a(n). a(4*n + 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 a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^9 (t / i)^9 f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 25 2007
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EXAMPLE
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G.f. = q + 16*q^2 + 256*q^4 - 1054*q^5 + 4096*q^8 + 6561*q^9 - 16864*q^10 - ...
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2]^18 (QPochhammer[ -q, q^2]^12 + 4 q / QPochhammer[ -q, q^2]^12), {q, 0, n}]; (* Michael Somos, Apr 12 2013 *)
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PROG
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(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 */
(PARI) {a(n) = local(A, p, e, x, y, z, a0, a1); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e = A[k, 2]; if( p==2, 16^e, if( p%4 == 3, if( e%2, 0, p^(4*e)), forstep( i=0, sqrtint(p), 2, if( issquare( p - i^2, &y), x = i; break)); a0 = 1; a1 = x = real( (x + I*y)^8 ) * 2; for( i=2, e, y = x*a1 - p^8*a0; a0=a1; a1=y); a1))))) }; /* Michael Somos, Nov 18 2014 */
(Sage) A = CuspForms( Gamma1(4), 9, prec=53).basis(); A[0] + 16*A[1]; # Michael Somos, Apr 12 2013
(Magma) A := Basis( CuspForms( Gamma1(4), 9), 50); A[1] + 16*A[2]; /* Michael Somos, Nov 16 2014 */
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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