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A002107
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Expansion of Product (1-x^k)^2, k=1..inf.
(Formerly M0091 N0028)
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5
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1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, 0, 0, 2, 3, -2, 2, 0, 0, -2, -2, 0, 0, -2, -1, 0, 2, 2, -2, 2, 1, 2, 0, 2, -2, -2, 2, 0, -2, 0, -4, 0, 0, 0, 1, -2, 0, 0, 2, 0, 2, 2, 1, -2, 0, 2, 2, 0, 0, -2, 0, -2, 0, -2, 2, 0, -4, 0, 0, -2, -1, 2, 0, 2, 0, 0, 0, -2, 2, 4, 1, 0, 0, 2, -2, 2, -2, 0, 0, 2, 0, -2, 0, -2, -2, 0, -2, 0, 0, 0, 2, -2, -1, -2, -2
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of partitions of n into an even number of distinct parts minus number of partitions of n into an odd number of distinct parts, with 2 types of each part. E.g. for n=4, we consider k and k* to be different versions of k and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*. The even partitions number 5 and the odd partitions number 4, so a(4)=5-4=1 - Jon Perry (perry(AT)globalnet.co.uk), Apr 04 2004
Also, number of different partitions of n into parts of -2 different kinds (based upon formal analogy) - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004
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REFERENCES
| M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. Soc., 21 (1889), 182-194.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
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
| T. D. Noe, Table of n, a(n) for n=0..1000
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
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FORMULA
| a(n)=b(12n+1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 7, 11 (mod 12), b(p^e) = (-1)^(e/2)(1+(-1)^e)/2 if p == 5 (mod 12), b(p^e) = (e+1)*(-1)^(e*x) if p == 1 (mod 12) and p = x^2+9y^2. - Michael Somos Sep 16 2006
a(0)=1, a(n)=1/n*sum(k=0,n-1, -2*a(k)*sigma_1(n-k)) - Joerg Arndt, Feb 5 2011
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PROG
| (PARI) {a(n)=local(A, p, e, x); if(n<0, 0, n=12*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p<5, 0, if(p%12>1, if(e%2, 0, (-1)^((p%12==5)*e/2)), for(i=1, sqrtint(p\9), if(issquare(p-9*i^2), x=i; break)); (e+1)*(-1)^(e*x))))))} /* Michael Somos Aug 30 2006 */
(PARI) {a(n)=if(n<0, 0, polcoeff( eta(x+x*O(x^n))^2, n))} /* Michael Somos Aug 30 2006 */
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CROSSREFS
| Cf. A000712 (reciprocal of g.f.).
Sequence in context: A063279 A124333 A144757 * A133099 A006571 A100889
Adjacent sequences: A002104 A002105 A002106 * A002108 A002109 A002110
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KEYWORD
| sign,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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