A000735 Expansion of Product_{k>=1} (1 - x^k)^12.

1, -12, 54, -88, -99, 540, -418, -648, 594, 836, 1056, -4104, -209, 4104, -594, 4256, -6480, -4752, -298, 5016, 17226, -12100, -5346, -1296, -9063, -7128, 19494, 29160, -10032, -7668, -34738, 8712, -22572, 21812, 49248, -46872, 67562, 2508, -47520, -76912, -25191, 67716

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice

Comments

Glaisher (1905, 1907) calls this sequence {Omega(m): m=1,3,5,7,9,11,...}. - N. J. A. Sloane, Nov 24 2018

Number 9 of the 74 eta-quotients listed in Table I of Martin (1996). See g.f. B(q) below: cusp form of weight 6 and level 4.

Grosswald uses b_n where b_{2n+1} = a(n).

Cynk and Hulek on page 14 in "The Example of Ahlgren" refer to a_p of the unique normalized weight 6 level 4 cusp form. - Michael Somos, Aug 24 2012

Expansion of q^(-1/2) * k(q) * k'(q)^4 * (K(q) / (Pi/2))^6 / 4 in powers of q where k(), k'(), K() are Jacobi elliptic functions. In Glaisher 1907 denoted by Omega(m) defined in section 62 on page 37. - Michael Somos, May 19 2013

Examples

			G.f. A(x) = 1 - 12*x + 54*x^2 - 88*x^3 - 99*x^4 + 540*x^5 - 418*x^6 - 648*x^7 + ...
G.f. B(q) = q - 12*q^3 + 54*q^5 - 88*q^7 - 99*q^9 + 540*q^11 - 418*q^13 - 648*q^15 + ...
		

References

• 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 p. 340.
• Glaisher, J. W. L. (1906). The arithmetical functions P(m), Q(m), Omega(m). Quart. J. Math, 37, 36-48.
• E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
• Newman, Morris; A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
• 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. Cynk and K. Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
• J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 5).
• Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
• Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
• K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Case k=12.
• Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
• Index entries for expansions of Product_{k >= 1} (1-x^k)^m
• Index entries for sequences mentioned by Glaisher

Crossrefs

Cf. A000145, A000729, A005758, A029751.

A209676 is the same except for signs.

This is a bisection of A227239.

Programs

Julia

# DedekindEta is defined in A000594.
A000735List(len) = DedekindEta(len, 12)
A000735List(42) |> println # Peter Luschny, Mar 10 2018

Magma

Basis( CuspForms( Gamma0(4), 6), 85) [1]; /* Michael Somos, Dec 09 2013 */

Maple

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-> -12): seq(a(n), n=0..45); # Alois P. Heinz, Sep 08 2008

Mathematica

CoefficientList[ Take[ Expand[ Product[(1 - x^k)^12, {k, 42}]], 42], x]
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^12, {q, 0, n}]; (* Michael Somos, May 19 2013 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - q^k, {k, n}]^12, {q, 0, n}]; (* Michael Somos, May 19 2013 *)

PARI

{a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^12, n))}; /* Michael Somos, Sep 21 2005 */

Sage

CuspForms( Gamma0(4), 6, prec=85).0; # Michael Somos, May 28 2013

Formula

Expansion of q^(-1/2) * eta(q)^12 in powers of q.

Euler transform of period 1 sequence [-12, ...]. - Michael Somos, Sep 21 2005

Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^4*w^2 + 48*(u*v*w)^2 + 4906*u^2*w^4 - u^6. - Michael Somos, Sep 21 2005

a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^5 * b(p^(e-2)). - Michael Somos, Mar 08 2006

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 64 (t/i)^6 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 24 2012

G.f.: (Product_{k>0} (1 - x^k))^12.

A000145(n) = A029751(n) + 16*a(n). - Michael Somos, Sep 21 2005

a(n) = (-1)^n * A209676(n).

Convolution inverse of A005758. Convolution square of A000729.

a(0) = 1, a(n) = -(12/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017

G.f.: exp(-12*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018