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Search: id:A107760
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| A107760 |
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Expansion of eta(q^3)eta(q^2)^6/(eta(q)^3 eta(q^6)^2) in powers of q. |
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5
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| 1, 3, 3, 3, 3, 0, 3, 6, 3, 3, 0, 0, 3, 6, 6, 0, 3, 0, 3, 6, 0, 6, 0, 0, 3, 3, 6, 3, 6, 0, 0, 6, 3, 0, 0, 0, 3, 6, 6, 6, 0, 0, 6, 6, 0, 0, 0, 0, 3, 9, 3, 0, 6, 0, 3, 0, 6, 6, 0, 0, 0, 6, 6, 6, 3, 0, 0, 6, 0, 0, 0, 0, 3, 6, 6, 3, 6, 0, 6, 6, 0, 3, 0, 0, 6, 0, 6, 0, 0, 0, 0, 12, 0, 6, 0, 0, 3, 6, 9, 0, 3, 0, 0, 6, 6
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.42).
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FORMULA
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Euler transform of period 6 sequence [3, -3, 2, -3, 3, -2, ...].
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)= +v^3 +u^2*w +4*v*w^2 -4*v^2*w -2*u*v*w.
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=(u1-u2)*(u1-u2-u3+u6) -3*u6*(u2-u6).
Expansion of psi(q)^3/psi(q^3) in powers of q where psi() is a Ramanujan theta function.
Expansion of (a(q) + a(q^2)) / 2 = b(q^2)^2 / b(q) in powers of q where a(), b() are cubic AGM functions. - Michael Somos, Aug 30 2008
Moebius transform is period 6 sequence [ 3, 0, 0, 0, -3, 0, ...]. - Michael Somos Aug 11 2009
a(n) = 3 * b(n) unless n=0 and b(n) is multiplicative with b(p^e) = 1 if p=2 or p=3; b(p^e) = 1+e if p == 1 (mod 6); b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6). - Michael Somos Aug 11 2009
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (27/4)^(1/2) (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A123330. - Michael Somos Aug 11 2009
G.f.: (Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k-1)))^3 / (Product_{k>0} (1 - x^(6*k)) / (1 - x^(6*k-3))). - Michael Somos Aug 11 2009
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EXAMPLE
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1 + 3*q + 3*q^2 + 3*q^3 + 3*q^4 + 3*q^6 + 6*q^7 + 3*q^8 + 3*q^9 + ... - Michael Somos Aug 11 2009
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PROGRAM
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(PARI) a(n)=if(n<1, n==0, 3*direuler(p=2, n, 1/(1-X)/(1-kronecker(-12, p)*X))[n])
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^3+A)*eta(x^2+A)^6/eta(x^6+A)^2/eta(x+A)^3, n))}
(PARI) {a(n)=if(n<1, n==0, 3*sumdiv(n, d, kronecker(-12, d)))}
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CROSSREFS
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a(n)=3*A035178(n), if n>0.
Sequence in context: A033700 A122916 A132973 this_sequence A138070 A081334 A106694
Adjacent sequences: A107757 A107758 A107759 this_sequence A107761 A107762 A107763
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KEYWORD
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nonn
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AUTHOR
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Michael Somos, May 24 2005
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