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A006922
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Expansion of 1/eta(q)^24; Fourier coefficients of T_{14}.
(Formerly M5160)
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11
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1, 24, 324, 3200, 25650, 176256, 1073720, 5930496, 30178575, 143184000, 639249300, 2705114880, 10914317934, 42189811200, 156883829400, 563116739584, 1956790259235, 6599620022400, 21651325216200
(list; graph; refs; listen; history; internal format)
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OFFSET
| -1,2
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COMMENTS
| Euler transform of period 1 sequence [24,24,...].
Equals A023021 convolved with A000041 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 09 2009]
Equals convolution square of A005758: (1, 12, 90, 520, 2535, 10908,...) [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 13 2009]
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REFERENCES
| C. J. Moreno and A. Rocha-Caridi, The exact formula for the weight multiplicities of affine Lie algebras, I, pp. 111-152 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
C. L. Siegel, Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bombay, 1980, pp. 249-268.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=-1..200
R. E. Borcherds, Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
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FORMULA
| G.f.: (1/x)(Product_{k>0} (1-x^k))^-24 = 1/\Delta (the discriminant in Siegel's notation.)
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EXAMPLE
| T_{14} = 1/q + 24 + 324q + 3200q^2 + 25650q^3 + ....
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MAPLE
| with (numtheory): b:= proc(n) option remember; `if`(n=0, 1, add (add (d*24, d=divisors(j)) *b(n-j), j=1..n)/n) end: a:= n->b(n+1): seq (a(n), n=-1..40); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 17 2008]
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MATHEMATICA
| max = 18; f[x_] := (1/x)*Product[1-x^k, {k, 1, max}]^-24; Join[{1}, CoefficientList[ Series[ f[x] - 1/x, {x, 0, max-1}], x]] (* From Jean-François Alcover, Oct 11 2011 *)
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PROG
| (PARI) a(n)=if(n<-1, 0, n++; polcoeff(eta(x+x*O(x^n))^-24, n))
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CROSSREFS
| Cf. A000594, A048057, A048100, A048101, A048110, A048145.
Cf. 24th column of A144064. [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 17 2008]
A023021, A000041 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 09 2009]
A005758 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 13 2009]
Sequence in context: A004413 A069779 A199301 * A036221 A022652 A138453
Adjacent sequences: A006919 A006920 A006921 * A006923 A006924 A006925
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Barry Brent (barryb(AT)primenet.com)
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