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A107653
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Expansion of q / (chi(q) * chi(q^3))^6 in powers of q where chi() is a Ramanujan theta function.
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3
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1, -6, 21, -68, 198, -510, 1248, -2904, 6393, -13604, 28044, -55956, 108982, -207552, 386622, -707216, 1271970, -2250582, 3925780, -6757272, 11483232, -19290824, 32057352, -52722744, 85884503, -138644292, 221885805, -352241792, 554892894
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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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LINKS
| Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Convolution inverse of A186829. - Michael Somos Feb 27 2011
Expansion of (eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2) * eta(q^6))^2)^6 in powers of q.
Euler transform of period 12 sequence [ -6, 6, -12, 0, -6, 12, -6, 0, -12, 6, -6, 0, ...]. - Michael Somos, Jun 13 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 pi i t). - Michael Somos Feb 27 2011
G.f.: x * (Product_{k>0} ((1 + x^(2*k)) * (1 + x^(6*k))) / ((1 + x^k) * (1 + x^(3*k))))^6 = x * (Product_{k>0} (1 + x^(2*k-1)) * (1 + x^(6*k-3)))^-6.
a(n) = -(-1)^n * A123653(n). - Michael Somos Feb 27 2011
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EXAMPLE
| q - 6*q^2 + 21*q^3 - 68*q^4 + 198*q^5 - 510*q^6 + 1248*q^7 + ...
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PROG
| (PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A)^2 * eta(x^6 + A)^2))^6, n))} /* Michael Somos Jun 13 2005 */
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CROSSREFS
| Cf. A123653.
Sequence in context: A119103 A180795 * A123653 A200761 A169687 A101904
Adjacent sequences: A107650 A107651 A107652 * A107654 A107655 A107656
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jun 07 2005
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EXTENSIONS
| Revised by Michael Somos, Jun 12 2005
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