A010816 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

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

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

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

EXAMPLE

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 + ...

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 *)

PROG

(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))};

(Julia) # DedekindEta is defined in A000594.

A010816List(len) = DedekindEta(len, 3)

A010816List(39) |> println # Peter Luschny, Mar 10 2018

(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

CROSSREFS