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A010816 Expansion of Product_{k>=1} (1 - x^k)^3.

%I A010816 #101 Jun 08 2025 15:19:47
%S A010816 1,-3,0,5,0,0,-7,0,0,0,9,0,0,0,0,-11,0,0,0,0,0,13,0,0,0,0,0,0,-15,0,0,
%T A010816 0,0,0,0,0,17,0,0,0,0,0,0,0,0,-19,0,0,0,0,0,0,0,0,0,21,0,0,0,0,0,0,0,
%U A010816 0,0,0,-23,0,0,0,0,0,0,0,0,0,0,0,25,0,0,0,0,0,0,0,0,0,0,0,0,-27,0,0,0,0,0,0,0
%N A010816 Expansion of Product_{k>=1} (1 - x^k)^3.
%C A010816 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A010816 Also, number of different partitions of n into parts of -3 different kinds (based upon formal analogy). - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004
%D A010816 T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 117, Problem 22.
%D A010816 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.14).
%D A010816 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003, p. 285, Theorem 357 (Jacobi).
%D A010816 D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 410, Problem 23.
%D A010816 S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 267 MR0099904 (20 #6340)
%H A010816 Seiichi Manyama, <a href="/A010816/b010816.txt">Table of n, a(n) for n = 0..10000</a>
%H A010816 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 A010816 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 A010816 A. Fink, R. K. Guy and M. Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics,  Vol 3, No 2 (2008).
%H A010816 V. Kotesovec, <a href="http://oeis.org/A258232/a258232_2.pdf">The integration of q-series</a>
%H A010816 M. Newman, <a href="http://dx.doi.org/10.1016/S1385-7258(56)50028-5">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.
%H A010816 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A010816 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A010816 <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F A010816 G.f.: Product_{k>=1} (1-x^k)^3 = Sum_{n>=0} (-1)^n*(2*n+1)*x^(n*(n+1)/2) (Jacobi).
%F A010816 Given g.f. A(x), then q * A(q^8) = eta(q^8)^3 = theta_2(q^4)*theta_3*(q^4)*theta_4(q^4) / 2 = theta_1'(q^4) / (2*Pi). - _Michael Somos_, Nov 08 2005
%F A010816 Given g.f. A(x), then x*A(x)^8 is g.f. for A000594.
%F A010816 a(n) = b(8*n + 1) where b() is multiplicative with b(p^e) = 0 if e odd, b(2^e) = 0^e, b(p^e) = p^(e/2) if p == 1 (mod 4), b(p^e) = (-p)^(e/2) if p == 3 (mod 4). - _Michael Somos_, Aug 22 2006
%F A010816 Expansion of f(-x)^3 in powers of x where f() is a Ramanujan theta function.
%F A010816 G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 2^(9/2) (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - _Michael Somos_, Sep 09 2007
%F A010816 a(3*n + 2) = a(5*n + 2) = a(5*n + 4) = a(9*n + 4) = a(9*n + 7) = 0. a(9*n + 1) = -3 * a(n). a(25*n + 3) = 5 * a(n). - _Michael Somos_, Sep 09 2007
%F A010816 a(3*n) = A116916(n).
%F A010816 a(n) = (t*(t+1)-2*n-1)*(t-r)*(-1)^(t+1), where t = floor(sqrt(2*(n+1))+1/2) and r = floor(sqrt(2*n)+1/2). - _Mikael Aaltonen_, Jan 17 2015
%F A010816 a(0) = 1, a(n) = -(3/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Mar 26 2017
%F A010816 G.f.: exp(-3*Sum_{k>=1} x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 05 2018
%F A010816 G.f.: Product_{n >= 1} (1 - q^(4*n))^3 * (1 + q^(4*n-1))^(-3) * (1 - q^(4*n-2))^6 * (1 + q^(4*n-3))^(-3). - _Peter Bala_, Jun 07 2025
%e A010816 G.f. = 1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 - 11*x^15 + 13*x^21 - 15*x^28 + ...
%e A010816 G.f. for b(n): = q - 3*q^9 + 5*q^25 - 7*q^49 + 9*q^81 - 11*q^121 + 13*q^169 + ...
%p A010816 S:= series(mul(1-x^k,k=1..200)^3,x,201):
%p A010816 seq(coeff(S,x,j),j=0..200); # _Robert Israel_, Feb 01 2018
%p A010816 A010816 := n -> if issqr(8*n+1) then isqrt(8*n+1); (-1)^iquo(%, 2) * % else 0 fi:
%p A010816 seq(A010816(n), n=0..98); # _Peter Luschny_, Apr 17 2022
%t A010816 a[ n_] := SeriesCoefficient[ EllipticThetaPrime[ 1, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}]; (* _Michael Somos_, Oct 22 2011 *)
%t A010816 a[ n_] := With[ {m = 8 n + 1}, If[m > 0 && OddQ[ Length @ Divisors @ m], Sqrt[m] KroneckerSymbol[-4, Sqrt[m]], 0]];  (* _Michael Somos_, Aug 26 2015 *)
%t A010816 CoefficientList[QPochhammer[q]^3 + O[q]^100, q] (* _Jean-François Alcover_, Nov 25 2015 *)
%t A010816 a[ n_] := With[ {x = Sqrt[8 n + 1]}, If[ IntegerQ[ x], (-1)^Quotient[ x, 2] x, 0]]; (* _Michael Somos_, Feb 01 2018 *)
%t A010816 a[ n_] := If[ n < 1, Boole[ n == 0], Times @@ (If[ # == 2 || OddQ[ #2], 0, (KroneckerSymbol[ -4, #] #)^(#2/2)] & @@@ FactorInteger[ 8 n + 1])]; (* _Michael Somos_, Feb 01 2018 *)
%o A010816 (PARI) {a(n) = my(x); if( n<0, 0, if( issquare( 8*n + 1, &x), (-1)^(x\2) * x))}; /* _Michael Somos_, Nov 08 2005 */
%o A010816 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3, n))};
%o A010816 (Julia) # DedekindEta is defined in A000594.
%o A010816 A010816List(len) = DedekindEta(len, 3)
%o A010816 A010816List(39) |> println # _Peter Luschny_, Mar 10 2018
%o A010816 (Python)
%o A010816 from sympy import integer_nthroot
%o A010816 def A010816(n):
%o A010816     a, b = integer_nthroot((n<<3)+1,2)
%o A010816     return (-a if a&2 else a) if b else 0 # _Chai Wah Wu_, Nov 02 2024
%Y A010816 Cf. A000594, A116916, A258407.
%K A010816 sign,easy,nice
%O A010816 0,2
%A A010816 _N. J. A. Sloane_