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Search: id:A106402
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| A106402 |
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Expansion of eta(q^3)^9/eta(q)^3 in powers of q. |
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1
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| 1, 3, 9, 13, 24, 27, 50, 51, 81, 72, 120, 117, 170, 150, 216, 205, 288, 243, 362, 312, 450, 360, 528, 459, 601, 510, 729, 650, 840, 648, 962, 819, 1080, 864, 1200, 1053, 1370, 1086, 1530, 1224, 1680, 1350, 1850, 1560, 1944, 1584, 2208, 1845, 2451, 1803
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Expansion of (c(q)/3)^3 in powers of q. Here c(q) is the third function in a cubic AGM analogue described by Borwein.
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REFERENCES
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J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 697.
G. E. Andrews, B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 314, Equ. (14.2.14).
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FORMULA
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Euler transform of period 3 sequence [3, 3, -6, ...].
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)=v^3+6uvw+8uw^2-u^2w.
G.f.: Sum_{k>0} k^2*x^k/(1+x^k+x^(2k)) = x Product_{k>0} (1-x^(3k))^9/(1-x^k)^3.
a(n) is multiplicative and a(p^e) = ((p^2)^(e+1)-u^(e+1))/(p^2-u) where u = 0, 1, -1 when p == 0, 1, 2 (mod 3). - Michael Somos Oct 19 2005
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EXAMPLE
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q + 3*q^2 + 9*q^3 + 13*q^4 + 24*q^5 + 27*q^6 + 50*q^7 + 51*q^8 +...
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PROGRAM
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(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x^3+A)^9/eta(x+A)^3, n))}
(PARI) a(n)=if(n<1, 0, sumdiv(n, d, d^2*kronecker(n/d, 3)))
(PARI) {a(n)=local(A, p, e, u); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; u=kronecker(-3, p); ((p^2)^(e+1)-u^(e+1))/(p^2-u))))}
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CROSSREFS
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Cf. A033687.
Sequence in context: A135370 A022408 A126827 this_sequence A125706 A113510 A163795
Adjacent sequences: A106399 A106400 A106401 this_sequence A106403 A106404 A106405
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KEYWORD
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nonn,mult
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AUTHOR
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Michael Somos, May 02 2005
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