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%I M4841 N2069 #103 Sep 08 2022 08:44:28
%S 1,-12,54,-88,-99,540,-418,-648,594,836,1056,-4104,-209,4104,-594,
%T 4256,-6480,-4752,-298,5016,17226,-12100,-5346,-1296,-9063,-7128,
%U 19494,29160,-10032,-7668,-34738,8712,-22572,21812,49248,-46872,67562,2508,-47520,-76912,-25191,67716
%N Expansion of Product_{k>=1} (1 - x^k)^12.
%C Glaisher (1905, 1907) calls this sequence {Omega(m): m=1,3,5,7,9,11,...}. - _N. J. A. Sloane_, Nov 24 2018
%C Number 9 of the 74 eta-quotients listed in Table I of Martin (1996). See g.f. B(q) below: cusp form of weight 6 and level 4.
%C Grosswald uses b_n where b_{2n+1} = a(n).
%C Cynk and Hulek on page 14 in "The Example of Ahlgren" refer to a_p of the unique normalized weight 6 level 4 cusp form. - _Michael Somos_, Aug 24 2012
%C Expansion of q^(-1/2) * k(q) * k'(q)^4 * (K(q) / (Pi/2))^6 / 4 in powers of q where k(), k'(), K() are Jacobi elliptic functions. In Glaisher 1907 denoted by Omega(m) defined in section 62 on page 37. - _Michael Somos_, May 19 2013
%D J. W. L. Glaisher, On the representations of a number as a sum of four squares, and on some allied arithmetical functions, Quarterly Journal of Pure and Applied Mathematics, 36 (1905), 305-358. See p. 340.
%D Glaisher, J. W. L. (1906). The arithmetical functions P(m), Q(m), Omega{m). Quart. J. Math, 37, 36-48.
%D E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
%D Newman, Morris; A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A000735/b000735.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H M. Boylan, <a href="http://dx.doi.org/10.1016/S0022-314X(02)00037-9">Exceptional congruences for the coefficients of certain eta-product newforms</a>, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
%H S. Cynk and K. Hulek, <a href="https://arxiv.org/abs/math/0509424">Construction and examples of higher-dimensional modular Calabi-Yau manifolds</a>, arXiv:math/0509424 [math.AG], 2005-2006.
%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 5).
%H Masao Koike, <a href="https://oeis.org/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H K. Ono, S. Robins and P. T. Wahl, <a href="http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/006.pdf">On the representation of integers as sums of triangular numbers</a>, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Case k=12.
%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>
%F Expansion of q^(-1/2) * eta(q)^12 in powers of q.
%F Euler transform of period 1 sequence [-12, ...]. - _Michael Somos_, Sep 21 2005
%F Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^4*w^2 + 48*(u*v*w)^2 + 4906*u^2*w^4 - u^6. - _Michael Somos_, Sep 21 2005
%F a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^5 * b(p^(e-2)). - _Michael Somos_, Mar 08 2006
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 64 (t/i)^6 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Aug 24 2012
%F G.f.: (Product_{k>0} (1 - x^k))^12.
%F A000145(n) = A029751(n) + 16*a(n). - _Michael Somos_, Sep 21 2005
%F a(n) = (-1)^n * A209676(n).
%F Convolution inverse of A005758. Convolution square of A000729.
%F a(0) = 1, a(n) = -(12/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 26 2017
%F G.f.: exp(-12*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 05 2018
%e G.f. A(x) = 1 - 12*x + 54*x^2 - 88*x^3 - 99*x^4 + 540*x^5 - 418*x^6 - 648*x^7 + ...
%e G.f. B(q) = q - 12*q^3 + 54*q^5 - 88*q^7 - 99*q^9 + 540*q^11 - 418*q^13 - 648*q^15 + ...
%p with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> -12): seq(a(n), n=0..45); # _Alois P. Heinz_, Sep 08 2008
%t CoefficientList[ Take[ Expand[ Product[(1 - x^k)^12, {k, 42}]], 42], x]
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q]^12, {q, 0, n}]; (* _Michael Somos_, May 19 2013 *)
%t a[ n_] := SeriesCoefficient[ Product[ 1 - q^k, {k, n}]^12, {q, 0, n}]; (* _Michael Somos_, May 19 2013 *)
%o (PARI) {a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^12, n))}; /* _Michael Somos_, Sep 21 2005 */
%o (Sage) CuspForms( Gamma0(4), 6, prec=85).0; # _Michael Somos_, May 28 2013
%o (Magma) Basis( CuspForms( Gamma0(4), 6), 85) [1]; /* _Michael Somos_, Dec 09 2013 */
%o (Julia) # DedekindEta is defined in A000594.
%o A000735List(len) = DedekindEta(len, 12)
%o A000735List(42) |> println # _Peter Luschny_, Mar 10 2018
%Y Cf. A000145, A000729, A005758, A029751.
%Y A209676 is the same except for signs.
%Y This is a bisection of A227239.
%K sign,easy,nice
%O 0,2
%A _N. J. A. Sloane_
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