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%I M4076 N1691 #109 Mar 20 2022 06:49:47
%S 1,-6,9,10,-30,0,11,42,0,-70,18,-54,49,90,0,-22,-60,0,-110,0,81,180,
%T -78,0,130,-198,0,-182,-30,90,121,84,0,0,210,0,-252,-102,-270,170,0,0,
%U -69,330,0,-38,420,0,-190,-390,0,-108,0,0,0,-300,99,442,210,0,418,-294,0,0,-510,378,-540,138,0
%N Expansion of Product_{k >= 1} (1 - x^k)^6.
%C This is Glaisher's function lambda(m). It appears to be defined only for odd m, and lambda(4t-1) = 0 (t >= 1), lambda(4t+1) = a(t) (t >= 0). - _N. J. A. Sloane_, Nov 25 2018
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Number 36 of the 74 eta-quotients listed in Table I of Martin (1996).
%C Dickson, v.2, p. 295 briefly states a result of Glaisher, 1883, pp 212-215. This result is that a(n) is the sum over all solutions of 16*n + 4 = x^2 + y^2 + z^2 + w^2 in nonnegative odd integers of chi(x) and is also the sum over all solutions of 8*n + 2 = x^2 + y^2 in nonnegative odd integers of chi(x) * chi(y) where chi(x) = x if x == 1 (mod 4) and -x if x == 3 (mod 4). [_Michael Somos_, Jun 18 2012]
%C Denoted by g_3(q) in Cynk and Hulek on page 8 as the unique weight 3 Hecke eigenform of level 16 with complex multiplication by i. - _Michael Somos_, Aug 24 2012
%C This is a member of an infinite family of integer weight modular forms. g_1 = A008441, g_2 = A002171, g_3 = A000729, g_4 = A215601, g_5 = A215472. - _Michael Somos_, Aug 24 2012
%D L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 295, and vol. 3, p. 134.
%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 page 340.
%D J. W. L. Glaisher, The arithmetical functions P(m), Q(m), Omega{m), Quart. J. Math, 37 (1906), 36-48.
%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 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 Seiichi Manyama, <a href="/A000729/b000729.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. MR1955423 (2003k:11071)
%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. Cynk and K. Hulek, <a href="http://arXiv.org/abs/math/0509424">Construction and examples of higher-dimensional modular Calabi-Yau manifolds</a>, arXiv:math/0509424 [math.AG], 2005-2006.
%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 J. W. L. Glaisher, <a href="http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0019%7CLOG_0034">Note on the Compositions of a Number as a Sum of Two and Four Uneven Squares</a>, Quarterly Journal of Pure and Applied Mathematics, 19 (1883), 212-215.
%H J. W. L. Glaisher, <a href="http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=PPN600494829_0020&DMDID=DMDLOG_0012">On the function chi(n)</a>, Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167.
%H J. W. L. Glaisher, <a href="/A002171/a002171.pdf">On the function chi(n)</a>, Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. [Annotated scanned copy]
%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 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 S. Milne and V. Leininger, <a href="https://www.intlpress.com/site/pub/files/_fulltext/journals/maa/1999/0006/0002/MAA-1999-0006-0002-a007.pdf">Some new infinite families of eta function identities</a>, Methods and Applications of Analysis 6 (1999), 225--248.
%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 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</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^(-1/4)/16 * theta_2(q)^4 * theta_3(q) * theta_4(q) in powers of q. - [Dickson, v. 3, p. 134] from Stieltjes footnote 160. _Michael Somos_, Jun 18 2012
%F Expansion of q^(-1/2) / 4 * k * k' * (K / (pi/2))^3 in powers of q^2 where k, k', K are Jacobi elliptic functions. - _Michael Somos_, Jun 22 2012
%F G.f.: Product_{k>0}(1 - x^k)^6.
%F Given g.f. A(x), then A(q^4) = f(-q^4)^6 = phi(q) * phi(-q) * psi(q^2)^4 where phi(), psi(), f() are Ramanujan theta functions. - _Michael Somos_, Aug 23 2006
%F a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - b(p^(e-2)) * p^2 if p == 1 (mod 4) and b(p) = 2 * (x^2 - y^2) where p = x^2 + y^2 and y is even. - _Michael Somos_, Aug 23 2006
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 64 (t/i)^3 f(t) where q = exp(2 Pi i t). - _Michael Somos_, Aug 24 2012
%F G.f.: Sum_{k>=0} a(k) * x^(4*k + 1) = (1/2) * Sum_{u,v in Z} (u*u - 4*v*v) * x^(u*u + 4*v*v). - _Michael Somos_, Jun 14 2007
%F G.f.: eta(x)^6 = Sum_{n>=0} (1+2n)^2*x^(n^2+n) + 2*Sum_{n>=0,k>=1} (1 + 4(n^2+n-k^2))*x^(n^2+n+k^2) - from the Milne and Leininger reference. [_Paul D. Hanna_, Mar 15 2010]
%F a(0) = 1, a(n) = -(6/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 26 2017
%F G.f.: exp(-6*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 05 2018
%F Let M be a positive integer whose prime factors are all congruent to 3 (mod 4) - see A004614. Then a( M^2*n + (M^2 - 1)/4 ) = M^2*a(n). See Cooper et al., equation 5. - _Peter Bala_, Dec 01 2020
%F a(n) = b(4*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = p^e * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = ((x+y*i)^(2*e+2) - (x-y*i)^(2*e+2))/((x+y*i)^2 - (x-y*i)^2) if p == 1 (mod 4) where p = x^2 + y^2 and x is odd. - _Jianing Song_, Mar 19 2022
%e G.f. = 1 - 6*x + 9*x^2 + 10*x^3 - 30*x^4 + 11*x^6 + 42*x^7 - 70*x^9 + 18*x^10 + ...
%e G.f. = q - 6*q^5 + 9*q^9 + 10*q^13 - 30*q^17 + 11*q^25 + 42*q^29 - 70*q^37 + ...
%t a[ n_] := SeriesCoefficient[ 1/16 EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q]^4 EllipticTheta[ 3, 0, q], {q, 0, 4 n + 1}]; (* _Michael Somos_, Jun 18 2012 *)
%t a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[ 16 n + 4]}, SeriesCoefficient[ Sum[ Mod[k, 2] q^k^2, {k, m}]^3 Sum[ KroneckerSymbol[ -4, k] k q^k^2, {k, m}], {q, 0, 16 n + 4}]]]; (* _Michael Somos_, Jun 12 2012 *)
%t a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[(1 - m) m ] (EllipticK[m] 2/Pi)^3 / (4 q^(1/2)), {q, 0, 2 n}]]; (* _Michael Somos_, Jun 22 2012 *)
%t a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^6, {x, 0, n}]; (* _Michael Somos_, May 17 2015 *)
%t a[ n_] := SeriesCoefficient[ QPochhammer[ x]^6, {x, 0, n}]; (* _Michael Somos_, May 17 2015 *)
%t a[ n_] := SeriesCoefficient[ (-1/4) EllipticThetaPrime[ 1, -Pi/4, q] EllipticTheta[ 1, -Pi/4, q]^3, {q, 0, 4 n + 1}]; (* _Michael Somos_, May 17 2015 *)
%t a[ n_] := SeriesCoefficient[ (-1/16) EllipticThetaPrime[ 1, 0, q] EllipticTheta[ 1, -Pi/2, q]^3, {q, 0, 4 n + 1}]; (* _Michael Somos_, May 17 2015 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^6, n))};
%o (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 4*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%4==3, if( e%2, 0, p^e), forstep( i=1, sqrtint(p), 2, if( issquare( p - i^2, &y), x=i; break)); a0=1; a1 = y = 2*(x^2 - y^2); for( i=2, e, x = y*a1 - p^2*a0; a0=a1; a1=x); a1)))}; /* _Michael Somos_, Aug 21 2006 */
%o (PARI) {a(n)=local(tn=(sqrtint(8*n+1)+1)\2);polcoeff(sum(m=0,tn,(1+2*m)^2*x^(m^2+m)+x*O(x^n)) + 2*sum(m=0,tn,sum(k=1,tn,(1+4*(m^2+m-k^2))*x^(m^2+m+k^2)+x*O(x^n))),n)} /* _Paul D. Hanna_, Mar 15 2010 */
%o (Magma) A := Basis( ModularForms( Gamma1(16), 3), 274); A[2] - 6*A[6] + 9*A[10] + 10*A[14] - 30*A[18]; /* _Michael Somos_, May 17 2015 */
%o (Magma) A := Basis( CuspForms( Gamma1(16), 3), 274); A[1] - 6*A[5]; /* _Michael Somos_, Jan 09 2017 */
%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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