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A000729
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Expansion of Product (1-x^k)^6, k=1..inf.
(Formerly M4076 N1691)
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1
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1, -6, 9, 10, -30, 0, 11, 42, 0, -70, 18, -54, 49, 90, 0, -22, -60, 0, -110, 0, 81, 180, -78, 0, 130, -198, 0, -182, -30, 90, 121, 84, 0, 0, 210, 0, -252, -102, -270, 170, 0, 0, -69, 330, 0, -38, 420, 0, -190, -390, 0, -108, 0, 0, 0, -300, 99, 442, 210, 0, 418, -294, 0, 0, -510, 378, -540, 138, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
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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.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 134.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
Newman, Morris; A table of the coefficients of the powers of $\eta(\tau)$. Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
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
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
S. Milne and V. Leininger, Some new infinite families of eta function identities, Methods and Applications of Analysis 6 (1999), 225--248. [From Paul D. Hanna (pauldhanna(AT)juno.com), Mar 15 2010]
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
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FORMULA
| G.f.: Product_{k>0}(1-x^k)^6.
Given g.f. A(x), then A(q^4)=f(-q^4)^6=phi(q)phi(-q)psi(q^2)^4 where phi(),psi(),f() are Ramanujan theta functions. - Michael Somos Aug 23 2006
a(n) = b(4n+1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^e(1+(-1)^e)/2 if p == 3 (mod 4), b(p^e)=b(p)b(p^(e-1))-b(p^(e-2))p^2 if p == 1 (mod 4) and b(p) = (-1)^y(x^2-y^2) where p = x^2+y^2. - Michael Somos Aug 23 2006
G.f.: Sum_{k>=0} a(k)*x^(4*k+1) = (1/2)* Sum_{u,v} (u*u -4*v*v)* x^(u*u +4*v*v). - Michael Somos Jun 14 2007
G.f.: eta(x)^6 = Sum_{n>=0} (1+2n)^2*x^(n^2+n) + 2*Sum_{n>=0,k>=1} (1 + 4(n^2+n-k^2))*x^(n^2+n+k^2) - from the Milne and Leininger reference. [From Paul D. Hanna (pauldhanna(AT)juno.com), Mar 15 2010]
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EXAMPLE
| q - 6*q^5 + 9*q^9 + 10*q^13 - 30*q^17 + 11*q^25 + 42*q^29 - 70*q^37 + ...
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PROG
| (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^6, n))}
(PARI) {a(n)= local(A, p, e, x, y, a0, a1); if(n<0, 0, n=4*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 0, if(p%4==3, if(e%2, 0, p^e), for(i=1, sqrtint(p\2), if(issquare(p-i^2, &y), x=i; break)); a0=1; a1=y=2*(x^2-y^2)*(-1)^y; for(i=2, e, x=y*a1-p^2*a0; a0=a1; a1=x); a1)))))} /* Michael Somos Aug 21 2006 */
(PARI) {a(n)=local(tn=(sqrtint(8*n+1)+1)\2); polcoeff(sum(m=0, tn, (1+2*m)^2*x^(m^2+m)+x*O(x^n)) + 2*sum(m=0, tn, sum(k=1, tn, (1+4*(m^2+m-k^2))*x^(m^2+m+k^2)+x*O(x^n))), n)} [From Paul D. Hanna (pauldhanna(AT)juno.com), Mar 15 2010]
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CROSSREFS
| Sequence in context: A130593 A051221 A029843 * A106248 A132725 A133451
Adjacent sequences: A000726 A000727 A000728 * A000730 A000731 A000732
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KEYWORD
| easy,sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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