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A093068
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Expansion of (eta(q^3)^2 * eta(q^7) * eta(q^63)) / (eta(q) * eta(q^9) * eta(q^21)^2) in powers of q.
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0
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1, 1, 2, 1, 3, 3, 4, 3, 5, 6, 9, 8, 11, 12, 15, 16, 18, 21, 26, 28, 35, 39, 47, 51, 58, 66, 77, 85, 97, 108, 125, 141, 156, 174, 195, 218, 245, 270, 304, 336, 377, 417, 467, 512, 573, 627, 702, 770, 853, 935, 1035, 1136, 1257, 1371, 1515, 1654, 1822, 1989, 2184, 2382
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Euler transform of period 63 sequence [ 1, 1, -1, 1, 1, -1, 0, 1, 0, 1, 1, -1, 1, 0, -1, 1, 1, 0, 1, 1, 0, 1, 1, -1, 1, 1, 0, 0, 1, -1, 1, 1, -1, 1, 0, 0, 1, 1, -1, 1, 1, 0, 1, 1, 0, 1, 1, -1, 0, 1, -1, 1, 1, 0, 1, 0, -1, 1, 1, -1, 1, 1, 0, ...].
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FORMULA
| Expansion of ( (c(q) * b(q^3) * b(q^7) * c(q^21)) / (b(q) * c(q^3) * c(q^7) * b(q^21)) )^(1/4) in powers of q where b(), c() are cubic AGM functions.
G.f. A(x) = B(x) / B(x^7) where B() is the g.f. of A112194.
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) = f(1/A(x), 1/A(x^2)) where f(u, v) = u^3 + v^3 - 2*u*v*(u + v) - u^2*v^2 - u*v.
G.f.: x * Product_{k>0} (1 - x^(3*k))^2 * (1 - x^(7*k)) * (1 - x^(63*k)) / ( (1 - x^k) * (1 - x^(9*k)) * (1 - x^(21*k))^2 ).
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EXAMPLE
| q + q^2 + 2*q^3 + q^4 + 3*q^5 + 3*q^6 + 4*q^7 + 3*q^8 + 5*q^9 + 6*q^10 + 9*q^11 + ...
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PROG
| (PARI) {a(n) = local(A, u, v); if( n<0, 0, A = x; for( k=2, n, u = A + x * O(x^k); v = subst(u, x, x^2); A -= x^k * polcoeff( u^3 + v^3 - 2*u*v*(u + v) - u^2*v^2 - u*v, k+2) / 2); polcoeff(A, n))}
(PARI) {a(n) = local(A); if( n<1, 0 , n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^7 + A) * eta(x^63 + A) / (eta(x + A) * eta(x^9 + A) * eta(x^21 + A)^2), n))}
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CROSSREFS
| Cf. A112194.
Sequence in context: A163281 A116921 A173989 * A097357 A123621 A151662
Adjacent sequences: A093065 A093066 A093067 * A093069 A093070 A093071
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KEYWORD
| nonn
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AUTHOR
| Michael Somos, Mar 17 2004
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