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A050468
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Sum_{d|n, n/d=1 mod 4} d^4 - Sum_{d|n, n/d=3 mod 4} d^4.
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7
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1, 16, 80, 256, 626, 1280, 2400, 4096, 6481, 10016, 14640, 20480, 28562, 38400, 50080, 65536, 83522, 103696, 130320, 160256, 192000, 234240, 279840, 327680, 391251, 456992, 524960, 614400, 707282, 801280, 923520, 1048576, 1171200
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Multiplicative because it is the Dirichlet convolution of A000583 = n^4 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. Christian G. Bower (bowerc(AT)usa.net) May 17, 2005.
Called E'_4(n) by Hardy.
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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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FORMULA
| Expansion of theta_3(q^2) * (theta_2(q)^8 + 4 * theta_2(q^2)^8) / 256 in powers of q^2. - Michael Somos, Jan 15 2012
Expansion of eta(q^2)^2 * eta(q^4)^4 * (eta(q)^4 + 20 * eta(q^4)^8 /
eta(q)^4) in powers of q. - Michael Somos, Jan 14 2012
Expansion of x * phi(x)^2 * (psi(x)^8 + 4 * x * psi(x^2)^8) in powers of x where phi(), psi() are Ramanujan theta funtions. - Michael Somos, Jan 15 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) = ((p^4)^(e+1) -(-1)^(e+1)) / (p^4 + 1) if p == 3 (mod 4). - Michael Somos, Jan 14 2012
a(2*n + 1) = A204342(n).
G.f.: Sum_{n>=1} n^4*x^n/(1+x^(2*n)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 16 2002
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EXAMPLE
| x + 16*x^2 + 80*x^3 + 256*x^4 + 626*x^5 + 1280*x^6 + 2400*x^7 + 4096*x^8 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * (-1)^((n/d - 1)/2) * d^4))} /* Michael Somos, Sep 12 2005 */
(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^4 * kronecker( -4, n\d)))} /* Michael Somos, Jan 14 2012 */
{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^4 + A)^4 * (eta(x + A)^4 + 20 * x * eta(x^4 + A)^8 / eta(x + A)^4), n))} /* Michael Somos, Jan 14 2012 */
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CROSSREFS
| Cf. A050469, A050470, A050471, A204342.
Sequence in context: A111732 A008511 A130810 * A068778 A034570 A165963
Adjacent sequences: A050465 A050466 A050467 * A050469 A050470 A050471
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KEYWORD
| nonn,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 23 1999
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