A029751 -id:A029751 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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 + ...

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).

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

Cf. A000145, A000729, A005758, A029751.
A209676 is the same except for signs.
This is a bisection of A227239.

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

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

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

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 *)

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

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

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

A000145 Number of ways of writing n as a sum of 12 squares.

Original entry on oeis.org

1, 24, 264, 1760, 7944, 25872, 64416, 133056, 253704, 472760, 825264, 1297056, 1938336, 2963664, 4437312, 6091584, 8118024, 11368368, 15653352, 19822176, 24832944, 32826112, 42517728, 51425088, 61903776, 78146664, 98021616

Offset: 0

N. J. A. Sloane

Apparently 8 | a(n). - Alexander R. Povolotsky, Oct 01 2011

			G.f. = 1 + 24*x + 264*x^2 + 1760*x^3 + 7944*x^4 + 25872*x^5 + 64416*x^6 + 133056*x^7 + ...

E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 314.

T. D. Noe, Table of n, a(n) for n = 0..10000
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
Index entries for sequences related to sums of squares

Row d=12 of A122141 and of A319574, 12th column of A286815.

Cf. A000735, A029751.

A := Basis( ModularForms( Gamma0(4), 6), 25); A[1] + 24*A[2] + 264*A[3] + 1760*A[4]; /* Michael Somos, Aug 15 2015 */

(sum(x^(m^2),m=-10..10))^12; # gives g.f. for first 100 terms
t1:=(sum(x^(m^2), m=-n..n))^12; t2:=series(t1,x,n+1); t2[n+1]; # N. J. A. Sloane, Oct 01 2011
A000145list := proc(len) series(JacobiTheta3(0, x)^12, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end: A000145list(27); # Peter Luschny, Oct 02 2018

SquaresR[12,Range[0,30]] (* Harvey P. Dale, Sep 07 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^12, {q, 0, n}]; (* Michael Somos, Aug 15 2015 *)
nmax = 30; CoefficientList[Series[Product[(1 - x^(2*k))^12 * (1 + x^(2*k - 1))^24, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 10 2018 *)

{a(n) = if( n<0, 0, polcoeff( sum( k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n))^12, n))}; /* Michael Somos, Sep 21 2005 */

Expansion of eta(q^2)^60 / (eta(q) * eta(q^4))^24 in powers of q.
Euler transform of period 4 sequence [24, -36, 24, -12, ...]. - Michael Somos, Sep 21 2005
G.f.: (Sum_k x^k^2)^12 = theta_3(q)^12.
a(n) = A029751(n) + 16 * A000735(n). - Michael Somos, Sep 21 2005
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).
a(n) = (24/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

cp's OEIS Frontend

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

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