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

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, 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, 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

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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

Examples

			G.f. = 1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 - 11*x^15 + 13*x^21 - 15*x^28 + ...
G.f. for b(n): = q - 3*q^9 + 5*q^25 - 7*q^49 + 9*q^81 - 11*q^121 + 13*q^169 + ...
		

References

• T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 117, Problem 22.
• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.14).
• 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. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 410, Problem 23.
• S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 267 MR0099904 (20 #6340)

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
• S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
• A. Fink, R. K. Guy and M. Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).
• V. Kotesovec, The integration of q-series
• M. 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.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
• Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Crossrefs

Cf. A000594, A116916, A258407.

Programs

Julia

# DedekindEta is defined in A000594.
A010816List(len) = DedekindEta(len, 3)
A010816List(39) |> println # Peter Luschny, Mar 10 2018

Maple

S:= series(mul(1-x^k,k=1..200)^3,x,201):
seq(coeff(S,x,j),j=0..200); # Robert Israel, Feb 01 2018
A010816 := n -> if issqr(8*n+1) then isqrt(8*n+1); (-1)^iquo(%, 2) * % else 0 fi:
seq(A010816(n), n=0..98); # Peter Luschny, Apr 17 2022

Mathematica

a[ n_] := SeriesCoefficient[ EllipticThetaPrime[ 1, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}]; (* Michael Somos, Oct 22 2011 *)
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 *)
CoefficientList[QPochhammer[q]^3 + O[q]^100, q] (* Jean-François Alcover, Nov 25 2015 *)
a[ n_] := With[ {x = Sqrt[8 n + 1]}, If[ IntegerQ[ x], (-1)^Quotient[ x, 2] x, 0]]; (* Michael Somos, Feb 01 2018 *)
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 *)

PARI

{a(n) = my(x); if( n<0, 0, if( issquare( 8*n + 1, &x), (-1)^(x\2) * x))}; /* Michael Somos, Nov 08 2005 */

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3, n))};

Python

from sympy import integer_nthroot
def A010816(n):
    a, b = integer_nthroot((n<<3)+1,2)
    return (-a if a&2 else a) if b else 0 # Chai Wah Wu, Nov 02 2024

Formula

G.f.: Product_{k>=1} (1-x^k)^3 = Sum_{n>=0} (-1)^n*(2*n+1)*x^(n*(n+1)/2) (Jacobi).

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

Given g.f. A(x), then x*A(x)^8 is g.f. for A000594.

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

Expansion of f(-x)^3 in powers of x where f() is a Ramanujan theta function.

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

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

a(3*n) = A116916(n).

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

a(0) = 1, a(n) = -(3/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017

G.f.: exp(-3*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018

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