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A050456
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Sum_{d|n, d=1 mod 4} d^4 - Sum_{d|n, d=3 mod 4} d^4.
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2
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1, 1, -80, 1, 626, -80, -2400, 1, 6481, 626, -14640, -80, 28562, -2400, -50080, 1, 83522, 6481, -130320, 626, 192000, -14640, -279840, -80, 391251, 28562, -524960, -2400, 707282, -50080, -923520, 1, 1171200, 83522, -1502400, 6481, 1874162, -130320
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Multiplicative because it is the Inverse Moebius transform of [1 0 -3^4 0 5^4 0 -7^4 ...], which is multiplicative. Christian G. Bower (bowerc(AT)usa.net) May 18, 2005.
Called E_4(n) by Hardy.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
| E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 120.
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)
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LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of ( eta(q)^4 * eta(q^2)^2 * (5 * eta(q)^8 / eta(q^4)^4 + 64 * q * eta(q^4)^4 ) - 5) / 4 in powers of q. - Michael Somos, Jan 14 2012
Expansion of (phi(x)^2 * (5 * phi(-x)^8 + 64 * x * psi(-x)^8) - 5) / 4 in powers of x where phi(), psi() are Rmanujan theta functions. - Michael Somos, Jan 14 2012
a(n) is multiplicative with a(2^e) = 1, a(p^e) = ((p^4)^(e+1) - 1) /
(p^4 - 1) if p == 1 (mod 4), a(p^e) = (1 - (-p^4)^(e+1)) / (1 + p^4) if p == 3 (mod 4). - Michael Somos, Jan 14 2012
G.f.: Sum_{k>0} (-1)^(k-1) * (2*k - 1)^4 * x^(2*k - 1) / (1 - x^(2*k - 1)).
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EXAMPLE
| x + x^2 - 80*x^3 + x^4 + 626*x^5 - 80*x^6 - 2400*x^7 + x^8 + 6481*x^9 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (d%2) * (-1)^((d-1)/2) * d^4)) /* Michael Somos, Sep 12 2005 */
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^4 * kronecker( -4, d)))}
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A)^4 * eta(x^2 + A)^2 * (5 * eta(x + A)^8 / eta(x^4 + A)^4 + 64 * x * eta(x^4 + A)^4 ) - 5) / 4, n))} /* Michael Somos, Jan 14 2012 */
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CROSSREFS
| Sequence in context: A128472 A093404 A031136 * A107930 A033400 A207144
Adjacent sequences: A050453 A050454 A050455 * A050457 A050458 A050459
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KEYWORD
| sign,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 23 1999
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