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A010816
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Expansion of Product_{k>=1} (1 - x^k)^3.
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24
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1, -3, 0, 5, 0, 0, -7, 0, 0, 0, 9, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, -15, 0, 0, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0, -19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -27, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,2
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COMMENTS
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Also, number of different partitions of n into parts of -3 different kinds (based upon formal analogy). - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004
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REFERENCES
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T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 117, Problem 22.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.14).
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003, p. 285, Theorem 357 (Jacobi).
D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 410, Problem 23.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 267 MR0099904 (20 #6340)
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LINKS
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FORMULA
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G.f.: Product_{k>=1} (1-x^k)^3 = Sum_{n>=0} (-1)^n*(2*n+1)*x^(n*(n+1)/2) (Jacobi).
Given g.f. A(x), then q * A(q^8) = eta(q^8)^3 = theta_2(q^4)*theta_3*(q^4)*theta_4(q^4) / 2 = theta_1'(q^4) / (2*Pi). - Michael Somos, Nov 08 2005
Given g.f. A(x), then x*A(x)^8 is g.f. for A000594.
a(n) = b(8*n + 1) where b() is multiplicative with b(p^e) = 0 if e odd, b(2^e) = 0^e, b(p^e) = p^(e/2) if p == 1 (mod 4), b(p^e) = (-p)^(e/2) if p == 3 (mod 4). - Michael Somos, Aug 22 2006
Expansion of f(-x)^3 in powers of x where f() is a Ramanujan theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 2^(9/2) (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 09 2007
a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(9*n + 1) = -3 * a(n). a(25*n + 3) = 5 * a(n). - Michael Somos, Sep 09 2007
a(n) = (t*(t+1)-2*n-1)*(t-r)*(-1)^(t+1), where t = floor(sqrt(2*(n+1))+1/2) and r = floor(sqrt(2*n)+1/2). - Mikael Aaltonen, Jan 17 2015
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EXAMPLE
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G.f. = 1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 - 11*x^15 + 13*x^21 - 15*x^28 + ...
G.f. for b(n): = q - 3*q^9 + 5*q^25 - 7*q^49 + 9*q^81 - 11*q^121 + 13*q^169 + ...
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MAPLE
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S:= series(mul(1-x^k, k=1..200)^3, x, 201):
A010816 := n -> if issqr(8*n+1) then isqrt(8*n+1); (-1)^iquo(%, 2) * % else 0 fi:
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MATHEMATICA
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a[ n_] := SeriesCoefficient[ EllipticThetaPrime[ 1, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}]; (* Michael Somos, Oct 22 2011 *)
a[ n_] := With[ {m = 8 n + 1}, If[m > 0 && OddQ[ Length @ Divisors @ m], Sqrt[m] KroneckerSymbol[-4, Sqrt[m]], 0]]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := With[ {x = Sqrt[8 n + 1]}, If[ IntegerQ[ x], (-1)^Quotient[ x, 2] x, 0]]; (* Michael Somos, Feb 01 2018 *)
a[ n_] := If[ n < 1, Boole[ n == 0], Times @@ (If[ # == 2 || OddQ[ #2], 0, (KroneckerSymbol[ -4, #] #)^(#2/2)] & @@@ FactorInteger[ 8 n + 1])]; (* Michael Somos, Feb 01 2018 *)
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PROG
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(PARI) {a(n) = my(x); if( n<0, 0, if( issquare( 8*n + 1, &x), (-1)^(x\2) * x))}; /* Michael Somos, Nov 08 2005 */
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3, n))};
(Julia) # DedekindEta is defined in A000594.
A010816List(len) = DedekindEta(len, 3)
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CROSSREFS
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KEYWORD
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sign,easy,nice
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AUTHOR
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STATUS
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approved
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