%I M3204 N1296 #101 Sep 08 2022 08:44:28
%S 1,-4,2,8,-5,-4,-10,8,9,0,14,-16,-10,-4,0,-8,14,20,2,0,-11,20,-32,-16,
%T 0,-4,14,8,-9,20,26,0,2,-28,0,-16,16,-28,-22,0,14,16,0,40,0,-28,26,32,
%U -17,0,-32,-16,-22,0,-10,32,-34,-8,14,0,45,-4,38,8,0,0,-34,-8,38,0,-22,-56,2,-28,0,0,-10,20,64,-40,-20,44
%N Expansion of Product_{k >= 1} (1 - x^k)^4.
%C Number 51 of the 74 eta-quotients listed in Table I of Martin (1996).
%C Ramanujan (see the link, pp. 155 and 157 Nr. 23.) conjectured the expansion coefficients called Psi_4(n) of eta^4(6*z) in powers of q = exp(2*Pi*i*z), Im(z) > 0, where i is the imaginary unit. In the Finch link on p. 5, multiplicity is used and Psi_4(p^r), called f(p^r), is given (see also b(p^e) formula given by Michael Somos, Aug 23 2006). Mordell proved this conjecture on pp. 121-122 based on Klein-Fricke, Theorie der elliptischen Modulfunktionen, 1892, Band II, p. 374. The product formula for the Dirichlet series, Mordell, eq. (7) for a=2,is used to find Psi_4(n), called f_2(n), from f_2(p) for primes p. The primes p = 2 and 3 do not appear in the product. - _Wolfdieter Lang_, May 03 2016
%D Morris Newman, 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 J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 415. Exer. 47.2.
%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="/A000727/b000727.txt">Table of n, a(n) for n = 0..10000</a>
%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.
%H Amanda Clemm, <a href="http://www.mdpi.com/2227-7390/4/1/5">Modular Forms and Weierstrass Mock Modular Forms</a>, Mathematics, volume 4, issue 1, (2016).
%H S. Cooper, M. D. Hirschhorn and R. Lewis, <a href="https://doi.org/10.1023/A:1009827103485">Powers of Euler's Product and Related Identities</a>, The Ramanujan Journal, Vol. 4 (2), 137-155 (2000).
%H S. R. Finch, <a href="https://arxiv.org/abs/math/0701251">Powers of Euler's q-Series</a>, arXiv:math/0701251 [math.NT], 2007.
%H M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.
%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 Louis J. Mordell, <a href="http://www.archive.org/stream/proceedingsofcam1920191721camb#page/n133">On Mr. Ramanujan's empirical expansions of modular functions</a>, Proceedings of the Cambridge Philosophical Society 19 (1917), pp. 117-124.
%H M. Newman, <a href="/A000727/a000727.pdf">A table of the coefficients of the powers of eta(tau)</a>, Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper18/page1.htm">On certain Arithmetical Functions</a>, Trans. Cambridge Philos. Soc. 22 (1916) 159-184; also in Collected papers, eds. G. H. Hardy, P. V. Seshu Aiyar and B. M. Wilson, Chelsea publ. Comp., 1962 (reprint from CUP 1927), pp. 136-162, 340- 341.
%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 Robert M. Ziff, <a href="http://dx.doi.org/10.1088/0305-4470/28/5/013">On Cardy's formula for the critical crossing probability in 2d percolation</a>, J. Phys. A. 28, 1249-1255 (1995).
%H <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F Euler transform of period 1 sequence [-4, -4, ...]. - _Michael Somos_, Apr 02 2005
%F Given g.f. A(x), then B(q) = q * A(q^3)^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = w*u^2 - v^3 + 16 * u*w^2. - _Michael Somos_, Apr 02 2005
%F a(n) = b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), b(p) = 0 if p == 5 (mod 6), b(p) = 2*x where p = x^2 + 3*y^2 == 1 (mod 6) and x == 1 (mod 3). - _Michael Somos_, Aug 23 2006
%F Coefficients of L-series for elliptic curve "36a1": y^2 = x^3 + 1. - _Michael Somos_, Jul 01 2004
%F a(n) = (-1)^n * A187076(n). a(2*n + 1) = -4 * A187150(n). a(25*n + 9) = a(25*n + 14) = a(25*n + 19) = a(25*n + 24) = 0. a(25*n + 4) = -5 * a(n). Convolution inverse of A023003. Convolution square of A002107. Convolution square is A000731.
%F a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 26 2017
%F G.f.: exp(-4*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 05 2018
%F Let M = p_1*...*p_k be a positive integer whose prime factors p_i (not necessarily distinct) are all congruent to 5 (mod 6). Then a( M^2*n + (M^2 - 1)/6 ) = (-1)^k*M*a(n). See Cooper et al., equation 4. - _Peter Bala_, Dec 01 2020
%F a(n) = b(6*n + 1) where b() is multiplicative with b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 2 (mod 3), b(p^e) = (((x+sqrt(-3)*y)/2)^(e+1) - ((x-sqrt(-3)*y)/2)^(e+1))/x if p == 1 (mod 3) where p = x^2 + 3*y^2 and x == 1 (mod 3). - _Jianing Song_, Mar 19 2022
%e G.f. = 1 - 4*x + 2*x^2 + 8*x^3 - 5*x^4 - 4*x^5 - 10*x^6 + 8*x^7 + 9*x^8 + ...
%e G.f. = q - 4*q^7 + 2*q^13 + 8*q^19 - 5*q^25 - 4*q^31 - 10*q^37 + 8*q^43 + ...
%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-> -4): seq(a(n), n=0..81); # _Alois P. Heinz_, Sep 08 2008
%t etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; a = etr[-4&]; Table[a[n], {n, 0, 81}] (* _Jean-François Alcover_, Mar 10 2014, after _Alois P. Heinz_ *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^4, {x, 0, n}]; (* _Michael Somos_, Jun 12 2014 *)
%t nmax = 80; CoefficientList[Series[Sum[Sum[(-1)^(k+m) * (2*k+1) * q^(k*(k+1)/2 + m*(3*m-1)/2), {k, 0, nmax}], {m, -nmax, nmax}], {q, 0, nmax}], q] (* _Vaclav Kotesovec_, Dec 06 2015 *)
%o (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 6*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%6==5, if(e%2, 0, (-1)^(e/2) * p^(e/2)), for( y=1, sqrtint(p\3), if( issquare( p - 3*y^2, &x), break)); a0=1; if( x%3!=1, x=-x); a1 = x = 2*x; for( i=2, e, y = x*a1 - p*a0; a0=a1; a1=y); a1)))}; /* _Michael Somos_, Aug 23 2006 */
%o (PARI) {a(n) = if( n<0, 0, polcoeff(eta(x + x * O(x^n))^4, n))};
%o (PARI) {a(n) = if( n<0, 0, ellak( ellinit( [0, 0, 0, 0, 1], 1), 6*n + 1))}; /* _Michael Somos_, Jul 01 2004 */
%o (Sage) ModularForms( Gamma0(36), 2, prec=493).0; # _Michael Somos_, Jun 12 2014
%o (Magma) qEigenform( EllipticCurve( [0, 0, 0, 0, 1]), 493); /* _Michael Somos_, Jun 12 2014 */
%o (Magma) A := Basis( ModularForms( Gamma0(36), 2), 493); A[2] - 4*A[8]; /* _Michael Somos_, Jun 12 2014 */
%o (Magma) Basis( CuspForms( Gamma0(36), 2), 493)[1]; /* _Michael Somos_, May 17 2015 */
%o (Julia) # DedekindEta is defined in A000594.
%o L000727List(len) = DedekindEta(len, 4)
%o L000727List(82) |> println # _Peter Luschny_, Mar 09 2018
%o (Magma) Coefficients(&*[(1-x^m)^4:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; // _Vincenzo Librandi_, Mar 10 2018
%Y Cf. A000731, A002107, A023003, A187076, A187150, A258404, A280328.
%Y Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A002107, A010815 - A010840.
%K sign,easy
%O 0,2
%A _N. J. A. Sloane_
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