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Search: id:A131944
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| A131944 |
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Expansion of (1-eta(q)^3 * eta(q^2)^3/( eta(q^3) * eta(q^6)))/3 in powers of q. |
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+0
2
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| 1, 1, -5, 1, 6, -5, 8, 1, -23, 6, 12, -5, 14, 8, -30, 1, 18, -23, 20, 6, -40, 12, 24, -5, 31, 14, -77, 8, 30, -30, 32, 1, -60, 18, 48, -23, 38, 20, -70, 6, 42, -40, 44, 12, -138, 24, 48, -5, 57, 31, -90, 14, 54, -77, 72, 8, -100, 30, 60, -30, 62, 32, -184, 1, 84, -60, 68, 18, -120, 48, 72, -23, 74, 38, -155
(list; graph; listen)
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OFFSET
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1,3
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.65).
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FORMULA
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Expansion of (1-b(q)*b(q^2))/3 where b() is a cubic AGM function.
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 4- 3^(e+1), a(p^e) = (p^(e+1)-1)/(p-1) if p>3.
G.f.: (1- Product_{k>0} ((1-x^k)* (1-x^(2k)))^3/( (1-x^(3k))* (1-x^(6k))))/3.
G.f.: Sum_{k>0} (6k-1)* x^(6k-1)/( 1-x^(6k-1)) -2*(6k-5)* x^(6k-3)/( 1-x^(6k-3)) +(6k-5)* x^(6k-5)/( 1-x^(6k-5)).
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EXAMPLE
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q + q^2 - 5*q^3 + q^4 + 6*q^5 - 5*q^6 + 8*q^7 + q^8 - 23*q^9 + 6*q^10 +...
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PROGRAM
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(PARI) {a(n)= if(n<1, 0, sumdiv(n, d, d*((d%6==1)+ (d%6==5)- 2*(d%6==3))))}
(PARI) {a(n)= local(A); if(n<0, 0, A= x*O(x^n); polcoeff( (1- eta(x+A)^3* eta(x^2+A)^3/ eta(x^3+A)/ eta(x^6+A))/3, n))}
(PARI) {a(n)= local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 1, if(p==3, 4-p^(e+1), (p^(e+1)-1)/(p-1))))))}
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CROSSREFS
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A131943(n)= -3*a(n) unless n=0.
Sequence in context: A088401 A077491 A086231 this_sequence A058651 A007397 A052345
Adjacent sequences: A131941 A131942 A131943 this_sequence A131945 A131946 A131947
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Jul 30 2007
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