A035016 -id:A035016 - cp's OEIS frontend

A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.

1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400, 181680, 272160, 319680, 490560, 527520, 743040, 846720, 1123440, 1179360, 1635120, 1646400, 2207520, 2311680, 2877120, 2920320, 3931200, 3780240, 4747680, 4905600, 6026880

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
E_8 is also the Barnes-Wall lattice in 8 dimensions.
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
The E_8 lattice is integral, unimodular, and even. The 240 shortest nonzero vectors in the lattice have norm squared 2. Of these vectors, 128 are all half-integer, and 112 are all integer. - Michael Somos, Jun 10 2019

			G.f. = 1 + 240*x + 2160*x^2 + 6720*x^3 + 17520*x^4 + 30240*x^5 + 60480*x^6 + ...
G.f. = 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + 30240*q^10 + 60480*q^12 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from N. J. A. Sloane)
D. Bump, Automorphic Forms and Representations, Cambr. Univ. Press, 1997, p. 29.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
Heng Huat Chan, Shaun Cooper, and Pee Choon Toh, Ramanujan's Eisenstein series and powers of Dedekind's eta-function, Journal of the London Mathematical Society 75.1 (2007): 225-242. See Q(q).
Henry Cohn and Stephen D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 and 24, arXiv:1603.04759 [math.MG], 2016.
H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561-578; reprinted in "Twelve Geometric Essays", pp. 20-39.
D. de Laat and F. Vallentin, A Breakthrough in Sphere Packing: The Search for Magic Functions, arXiv preprint arXiv:1607.02111 [math.MG], 2016.
Yang-Hui He and John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998
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.]
Robert V. Moody and Jiri Patera, Fast recursion formula for weight multiplicities, Bulletin of the American Mathematical Society 7.1 (1982): 237-242.
G. Nebe and N. J. A. Sloane, Home page for E_8 lattice
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.
S. Ramanujan, On the coefficients in the expansions of certain modular functions, Proc. Royal Soc., A, 95 (1918), 144-155.
Michael Somos, Introduction to Ramanujan theta functions
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Seven Staggering Sequences.
Maryna S. Viazovska, The sphere packing problem in dimension 8, arXiv preprint arXiv:1603.04246 [math.NT], 2016.
Martin H. Weissman, Octonions, Cubes, Embeddings, March 2, 2009.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Eisenstein Series.
Eric Weisstein's World of Mathematics, Leech Lattice.
Eric Weisstein's World of Mathematics, Barnes-Wall Lattice
Wikipedia, Eisenstein series
Wikipedia, E_8 lattice
Index entries for sequences related to Eisenstein series
Index entries for sequences related to Barnes-Wall lattices

Cf. A046948 (partial sums), A000143, A108091 (eighth root).
Cf. A008410, A008411, A001158.
Cf. A006352 (E_2), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).
Cf. A007331 (theta_2(q)^8 / 256), A000143 (theta_3(q)^8), A035016 (theta_4(q)^8).

Basis( ModularForms( Gamma1(1), 4), 29) [1]; /* Michael Somos, May 11 2015 */

L := Lattice("E",8); A := ThetaSeries(L, 57); A; /* Michael Somos, Jun 10 2019 */

with(numtheory); E := proc(k) local n,t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n,n=1..60); series(t1,q,60); end; E(4);

a[ n_] := If[ n < 1, Boole[n == 0], 240 DivisorSigma[ 3, n]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^2 + 14 t2 t3 + t3^2], {q, 0, n}]; (* Michael Somos, Jun 04 2014 *)
max = 30; s = 1 + 240*Sum[k^3*(q^k/(1 - q^k)), {k, 1, max}] + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, after Gene Ward Smith *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^2 - t2 t3 + t3^2], {q, 0, 2 n}]; (* Michael Somos, Jul 31 2016 *)

{a(n) = if( n<1, n==0, 240 * sigma(n, 3))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^24 + 256 * x * eta(x^2 + A)^24) / (eta(x + A) * eta(x^2 + A))^8, n))}; /* Michael Somos, Dec 30 2008 */

q='q+O('q^50); Vec((eta(q)^24+256*q*eta(q^2)^24)/(eta(q)*eta(q^2))^8) \\ Altug Alkan, Sep 30 2018

from sympy import divisor_sigma
def a(n): return 1 if n == 0 else 240 * divisor_sigma(n, 3)
[a(n) for n in range(51)]  # Indranil Ghosh, Jul 15 2017

ModularForms(Gamma1(1), 4, prec=30).0 ; # Michael Somos, Jun 04 2013

Can also be expressed as E4(q) = 1 + 240*Sum_{i >= 1} i^3 q^i/(1 - q^i) - Gene Ward Smith, Aug 22 2006
Theta series of E_8 lattice = 1 + 240 * Sum_{m >= 1} sigma_3(m) * q^(2*m), where sigma_3(m) is the sum of the cubes of the divisors of m (A001158).
Expansion of (phi(-q)^8 - (2 * phi(-q) * phi(q))^4 + 16 * phi(q)^8) in powers of q where phi() is a Ramanujan theta function.
Expansion of (eta(q)^24 + 256 * eta(q^2)^24) / (eta(q) * eta(q^2))^8 in powers of q. - Michael Somos, Dec 30 2008
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 33*v^2 + 256*w^2 - 18*u*v + 16*u*w - 288*v*w . - Michael Somos, Jan 05 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 + 16*u2^2 + 81*u3^2 + 1296*u6^2 - 14*u1*u2 - 18*u1*u3 + 30*u1*u6 + 30*u2*u3 - 288*u2*u6 - 1134*u3*u6 . - Michael Somos, Apr 15 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = u^3*v + 9*w*u^3 - 84*u^2*v^2 + 246*u*v^3 - 253*v^4 - 675*w*u^2*v + 729*w^2*u^2 - 4590*w*u*v^2 + 19926*w*v^3 - 54675*w^2*u*v + 59049*w^3*u + 531441*w^3*v - 551124*w^2*v^2 . - Michael Somos, Apr 15 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^4 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008
Convolution square is A008410. A008411 is convolution of this sequence with A008410.
Expansion of Ramanujan's function Q(q^2) = 12 (omega/Pi)^4 g2 (Weierstrass invariant) in powers of q^2.
Expansion of a(q) * (a(q)^3 + 8*c(q)^3) in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, Jan 14 2015
G.f. is (theta_2(q)^8 + theta_3(q)^8 + theta_4(q)^8) / 2 where q = exp(Pi i t). So a(n) = A008430(n) + 128*A007331(n) (= A000143(2*n) + 128*A007331(n) = A035016(2*n) + 128*A007331(n)). - Seiichi Manyama, Sep 30 2018
a(n) = 240*A001158(n) if n>0. - Michael Somos, Oct 01 2018
Sum_{k=1..n} a(k) ~ 2 * Pi^4 * n^4 / 3. - Vaclav Kotesovec, Jan 14 2024

A007331 Fourier coefficients of E_{infinity,4}.

Original entry on oeis.org

0, 1, 8, 28, 64, 126, 224, 344, 512, 757, 1008, 1332, 1792, 2198, 2752, 3528, 4096, 4914, 6056, 6860, 8064, 9632, 10656, 12168, 14336, 15751, 17584, 20440, 22016, 24390, 28224, 29792, 32768, 37296, 39312, 43344, 48448, 50654, 54880, 61544, 64512

Offset: 0

N. J. A. Sloane, Mira Bernstein

E_{infinity,4} is the unique normalized weight-4 modular form for Gamma_0(2) with simple zeros at i*infinity. Since this has level 2, it is not a cusp form, in contrast to A002408.
a(n+1) is the number of representations of n as a sum of 8 triangular numbers (from A000217). See the Ono et al. link, Theorem 5.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) gives the sum of cubes of divisors d of n such that n/d is odd. This is called sigma^#3(n) in the Ono et al. link. See a formula below. - _Wolfdieter Lang, Jan 12 2017

			G.f. = q + 8*q^2 + 28*q^3 + 64*q^4 + 126*q^5 + 224*q^6 + 344*q^7 + 512*q^8 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139, Ex (ii).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1001 from T. D. Noe)
B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 187.
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. 4 and p. 8).
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.]
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. Theorem 5.
H. Rosengren, Sums of triangular numbers from the Frobenius determinant, arXiv:math/0504272 [math.NT], 2005.
Index entries for sequences mentioned by Glaisher.

Cf. A002408, A004017, A035016, A045825, A076577, A096960, A222072.

Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809, A076577.

Basis( ModularForms( Gamma0(2), 4), 10) [2]; /* Michael Somos, May 27 2014 */

nmax:=40: seq(coeff(series(x*(product((1-x^k)^8*(1+x^k)^16, k=1..nmax)), x, n+1), x, n), n=0..nmax); # Vaclav Kotesovec, Oct 14 2015

Prepend[Table[Plus @@ (Select[Divisors[k + 1], OddQ[(k + 1)/#] &]^3), {k, 0, 39}], 0] (* Ant King, Dec 04 2010 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^8 / 256, {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ d^3 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Jun 04 2013 *)
f[n_] := Total[(2n/Select[ Divisors[ 2n], Mod[#, 4] == 2 &])^3]; Flatten[{0, Array[f, 40] }] (* Robert G. Wilson v, Mar 26 2015 *)
nmax=60; CoefficientList[Series[x*Product[(1-x^k)^8 * (1+x^k)^16, {k,1,nmax}],{x,0,nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
QP = QPochhammer; s = q * (QP[-1, q]/2)^16 * QP[q]^8 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^3))}; /* Michael Somos, May 31 2005 */

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^8, n))}; /* Michael Somos, May 31 2005 */

a(n)=my(e=valuation(n,2)); 8^e * sigma(n/2^e, 3) \\ Charles R Greathouse IV, Sep 09 2014

from sympy import divisors
def a(n):
    return 0 if n == 0 else sum(((n//d)%2)*d**3 for d in divisors(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 24 2017

ModularForms( Gamma0(2), 4, prec=33).1; # Michael Somos, Jun 04 2013

G.f.: q * Product_{k>=1} (1-q^k)^8 * (1+q^k)^16. - corrected by Vaclav Kotesovec, Oct 14 2015

a(n) = Sum_{0

G.f.: Sum_{n>0} n^3*x^n/(1-x^(2*n)). - Vladeta Jovovic, Oct 24 2002

Expansion of Jacobi theta constant theta_2(q)^8 / 256 in powers of q.

Expansion of eta(q^2)^16 / eta(q)^8 in powers of q. - Michael Somos, May 31 2005

Expansion of x * psi(x)^8 in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jan 15 2012

Expansion of (Q(x) - Q(x^2)) / 240 in powers of x where Q() is a Ramanujan Lambert series. - Michael Somos, Jan 15 2012

Expansion of E_{gamma,2}^2 * E_{0,4} in powers of q.

Euler transform of period 2 sequence [8, -8, ...]. - Michael Somos, May 31 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 - u^2*w + 16*u*v*w - 32*v^2*w + 256*v*w^2. - Michael Somos, May 31 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 16^(-1) (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A035016. - Michael Somos, Jan 11 2009

Multiplicative with a(2^e) = 2^(3e), a(p^e) = (p^(3(e+1))-1)/(p^3-1). - Mitch Harris, Jun 13 2005

Dirichlet convolution of A154955 by A001158. Dirichlet g.f. zeta(s)*zeta(s-3)*(1-1/2^s). - R. J. Mathar, Mar 31 2011

A002408(n) = -(-1)^n * a(n).

Convolution square of A008438. - Michael Somos, Jun 15 2014

a(1) = 1, a(n) = (8/(n-1))*Sum_{k=1..n-1} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017

Sum_{k=1..n} a(k) ~ c * n^4, where c = Pi^4/384 = 0.253669... (A222072). - Amiram Eldar, Oct 19 2022

Additional comments from Barry Brent (barryb(AT)primenet.com)
Wrong Maple program replaced by Vaclav Kotesovec, Oct 14 2015
a(0)=0 prepended by Vaclav Kotesovec, Oct 14 2015

A000143 Number of ways of writing n as a sum of 8 squares.

Original entry on oeis.org

1, 16, 112, 448, 1136, 2016, 3136, 5504, 9328, 12112, 14112, 21312, 31808, 35168, 38528, 56448, 74864, 78624, 84784, 109760, 143136, 154112, 149184, 194688, 261184, 252016, 246176, 327040, 390784, 390240, 395136, 476672, 599152, 596736, 550368, 693504, 859952

Offset: 0

N. J. A. Sloane

The relevant identity for the o.g.f. is theta_3(x)^8 = 1 + 16*Sum_{j >= 1} j^3*x^j/(1 - (-1)^j*x^j). See the Hardy-Wright reference, p. 315. - Wolfdieter Lang, Dec 08 2016

			1 + 16*q + 112*q^2 + 448*q^3 + 1136*q^4 + 2016*q^5 + 3136*q^6 + 5504*q^7 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61); p. 79 Eq. (32.32).
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, pp. 314 - 315.

T. D. Noe, Table of n, a(n) for n = 0..10000
J. M. Borwein and K.-K. S. Choi, On Dirichlet series for sums of squares, Ramanujan J. 7 (1-3) (2003) 95-127.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
P. J. C. Lamont, The number of Cayley integers of given norm, Proceedings of the Edinburgh Mathematical Society, 25.1 (1982): 101-103. See (5).
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
M. Peters, Sums of nine squares, Acta Arith., 102 (2002), 131-135.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to sums of squares

8th column of A286815. - Seiichi Manyama, May 27 2017
Row d=8 of A122141.
Cf. A008457, A035016, A010815.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cf. A004018, A000118, A000141 for the expansion of the powers of 2, 4, 6 of theta_3(x).

# JacobiTheta3 is defined in A000122.
A000143List(len) = JacobiTheta3(len, 8)
A000143List(37) |> println # Peter Luschny, Mar 12 2018

(sum(x^(m^2),m=-10..10))^8;
with(numtheory); rJ := n-> if n=0 then 1 else 16*add((-1)^(n+d)*d^3, d in divisors(n)); fi; [seq(rJ(n),n=0..100)]; # N. J. A. Sloane, Sep 15 2018

Table[SquaresR[8, n], {n, 0, 33}] (* Ray Chandler, Dec 06 2006 *)
SquaresR[8,Range[0,50]] (* Harvey P. Dale, Aug 26 2011 *)
QP = QPochhammer; s = (QP[q^2]^5/(QP[q]*QP[q^4])^2)^8 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

{a(n) = if( n<1, n==0, 16 * (-1)^n * sumdiv( n, d, (-1)^d * d^3))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2)^8, n))} /* Michael Somos, Sep 25 2005 */

from math import prod
from sympy import factorint
def A000143(n): return prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(n).items())<<4 if n else 1 # Chai Wah Wu, Jun 21 2024

Q = DiagonalQuadraticForm(ZZ, [1]*8)
Q.representation_number_list(60) # Peter Luschny, Jun 20 2014

Expansion of theta_3(z)^8. Also a(n)=16*(-1)^n*Sum_{0

Expansion of phi(q)^8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 21 2008

Expansion of (eta(q^2)^5 / (eta(q) * eta(q^4))^2)^8 in powers of q. - Michael Somos, Sep 25 2005

G.f.: s(2)^40/(s(1)*s(4))^16, where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]

Euler transform of period 4 sequence [16, -24, 16, -8, ...]. - Michael Somos, Apr 10 2005

a(n) = 16 * b(n) and b(n) is multiplicative with b(p^e) = (p^(3*e+3) - 1) / (p^3 - 1) -2[p<3]. - Michael Somos, Sep 25 2005

G.f.: 1 + 16 * Sum_{k>0} k^3 * x^k / (1 - (-x)^k). - Michael Somos, Sep 25 2005

A035016(n) = (-1)^n * a(n). 16 * A008457(n) = a(n) unless n=0.

Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = 16*(1 - 2^(1-s) + 4^(2-s))*zeta(s)*zeta(s-3). [Borwein and Choi], R. J. Mathar, Jul 02 2012

a(n) = (16/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Sum_{k=1..n} a(k) ~ Pi^4 * n^4 /24. - Vaclav Kotesovec, Jul 12 2024

A138503 a(n) = Sum_{d|n} (-1)^(d-1)*d^3.

Original entry on oeis.org

1, -7, 28, -71, 126, -196, 344, -583, 757, -882, 1332, -1988, 2198, -2408, 3528, -4679, 4914, -5299, 6860, -8946, 9632, -9324, 12168, -16324, 15751, -15386, 20440, -24424, 24390, -24696, 29792, -37447, 37296, -34398, 43344, -53747, 50654, -48020, 61544, -73458

Offset: 1

Michael Somos, Mar 21 2008

Also, expansion of (1 - phi(-q)^8) / 16 in powers of q where phi() is a Ramanujan theta function.

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

			G.f. = q - 7*q^2 + 28*q^3 - 71*q^4 + 126*q^5 - 196*q^6 + 344*q^7 - 583*q^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
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. 4 and p. 8).
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Index entries for sequences mentioned by Glaisher.

Divisor sums Sum_{d|n} (-1)^(d-1)*d^k: A048272 (k = 0), A002129 (k = 1), A321543 (k = 2), A279395 (k = 4, unsigned), A321544 - A321551 (k = 5 to k = 12).

Cf. A008457, A035016.

with(numtheory):
a := n -> add( (-1)^(d-1)*d^3, d in divisors(n) ): seq(a(n), n = 1..40);
#  Peter Bala, Jan 11 2021

a[ n_] := If[ n < 0, 0, DivisorSum[ n, -(-1)^# #^3&]]; (* Michael Somos, Sep 25 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, q]^8) / 16, {q, 0, n}]; (* Michael Somos, Sep 25 2015 *)
nmax = 40; Rest[CoefficientList[Series[-Product[((1-q^k)/(1+q^k))^8, {k, 1, nmax}]/16, {q, 0, nmax}], q]] (* Vaclav Kotesovec, Sep 26 2015 *)
f[p_, e_] := (p^(3*e + 3) - 1)/(p^3 - 1); f[2, e_] := 2 - (2^(3*e + 3) - 1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 04 2022 *)

{a(n) = if( n<0, 0, sumdiv(n, d, -(-1)^d * d^3))};

Expansion of (1 - (eta(q)^2 / eta(q^2))^8) / 16 in powers of q.
a(n) is multiplicative with a(2^e) = -(8^(e+1) - 15) / 7, a(p^e) = ((p^3)^(e+1) - 1) / (p^3 - 1).
G.f.: Sum_{k>0} k^3 * -(-x)^k / (1 - x^k).
a(n) = -(-1)^n * A008457(n). -16 * a(n) = A035016(n) unless n=0.
G.f.: Sum_{n >= 1} x^n*(1 - 4*x^n + x^(2*n))/(1 + x^n)^4. - Peter Bala, Jan 11 2021

Simpler definition from N. J. A. Sloane, Nov 23 2018

A092820 a(0) = 1; for n>0, a(n) = 16 times sum of cubes of divisors of n.

Original entry on oeis.org

1, 16, 144, 448, 1168, 2016, 4032, 5504, 9360, 12112, 18144, 21312, 32704, 35168, 49536, 56448, 74896, 78624, 109008, 109760, 147168, 154112, 191808, 194688, 262080, 252016, 316512, 327040, 401792, 390240, 508032, 476672, 599184, 596736, 707616, 693504

Offset: 0

N. J. A. Sloane, Sep 11 2004

nonn

Cf. A004009, A008457, A035016.

A319307 Expansion of theta_4(q)^16 in powers of q = exp(Pi i t).

Original entry on oeis.org

1, -32, 480, -4480, 29152, -140736, 525952, -1580800, 3994080, -8945824, 18626112, -36714624, 67978880, -118156480, 197120256, -321692928, 509145568, -772845120, 1143441760, -1681379200, 2428524096, -3392205824, 4658843520, -6411152640, 8705492608, -11488092896

Offset: 0

Seiichi Manyama, Sep 16 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

theta_4(q)^b: A002448 (b=1), A104794 (b=2), A213384 (b=3), A096727 (b=4), A035016 (b=8), A286346 (b=12), this sequence (b=16), A319308 (b=20), A319309 (b=24), A319310 (b=28).

Cf. A000152.

Expansion of eta(q)^32 / eta(q^2)^16 in powers of q.

A319308 Expansion of theta_4(q)^20 in powers of q = exp(Pi i t).

Original entry on oeis.org

1, -40, 760, -9120, 77560, -497648, 2508000, -10232640, 34729720, -100906760, 259114704, -606957280, 1327461600, -2738111280, 5341699520, -9915552192, 17701924600, -30615844560, 51294999960, -83279292960, 131880275664, -204949382400, 312126610080, -464844224960, 680432137440

Offset: 0

Seiichi Manyama, Sep 16 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

theta_4(q)^b: A002448 (b=1), A104794 (b=2), A213384 (b=3), A096727 (b=4), A035016 (b=8), A286346 (b=12), A319307 (b=16), this sequence (b=20), A319309 (b=24), A319310 (b=28).

Expansion of eta(q)^40 / eta(q^2)^20 in powers of q.

A319309 Expansion of theta_4(q)^24 in powers of q = exp(Pi i t).

Original entry on oeis.org

1, -48, 1104, -16192, 170064, -1362336, 8662720, -44981376, 195082320, -721175536, 2319457632, -6631997376, 17231109824, -41469483552, 93703589760, -200343312768, 407488018512, -793229226336, 1487286966928, -2697825744960, 4744779429216, -8110465650176

Offset: 0

Seiichi Manyama, Sep 16 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

theta_4(q)^b: A002448 (b=1), A104794 (b=2), A213384 (b=3), A096727 (b=4), A035016 (b=8), A286346 (b=12), A319307 (b=16), A319308 (b=20), this sequence (b=24), A319310 (b=28).

Cf. A000156.

Expansion of eta(q)^48 / eta(q^2)^24 in powers of q.

A319310 Expansion of theta_4(q)^28 in powers of q = exp(Pi i t).

Original entry on oeis.org

1, -56, 1512, -26208, 327656, -3147984, 24189984, -152867520, 811401192, -3681079640, 14500933104, -50376047904, 156797510688, -444306558864, 1163495873088, -2851049839680, 6597606440936, -14512424533488, 30505974273096, -61591664700384, 119983597365744, -226303038736128

Offset: 0

Seiichi Manyama, Sep 16 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

theta_4(q)^b: A002448 (b=1), A104794 (b=2), A213384 (b=3), A096727 (b=4), A035016 (b=8), A286346 (b=12), A319307 (b=16), A319308 (b=20), A319309 (b=24), this sequence (b=28).

Expansion of eta(q)^56 / eta(q^2)^28 in powers of q.

A319553 Expansion of 1/theta_4(q)^8 in powers of q = exp(Pi i t).

Original entry on oeis.org

1, 16, 144, 960, 5264, 25056, 106944, 418176, 1520784, 5201232, 16871648, 52252992, 155341248, 445226848, 1234726272, 3323392128, 8704504976, 22234655520, 55498917840, 135595345600, 324759439584, 763505859072, 1764050361152, 4009763323008, 8975341703616, 19800832628336

Offset: 0

Seiichi Manyama, Sep 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

1/theta_4(q)^b: A015128 (b=1), A001934 (b=2), A319552 (b=3), A284286 (b=4), this sequence (b=8), A319554 (b=12).

Cf. A002131, A002448 (theta_4(q)), A004409, A035016.

N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^(2*k))/(1-x^k)^2)^8))

Convolution inverse of A035016.
a(n) = (-1)^n * A004409(n).
a(0) = 1, a(n) = (16/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0.
G.f.: Product_{k>=1} ((1 - x^(2k))/(1 - x^k)^2)^8.

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A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.

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A007331 Fourier coefficients of E_{infinity,4}.

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A000143 Number of ways of writing n as a sum of 8 squares.

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A138503 a(n) = Sum_{d|n} (-1)^(d-1)*d^3.

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A092820 a(0) = 1; for n>0, a(n) = 16 times sum of cubes of divisors of n.

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A319307 Expansion of theta_4(q)^16 in powers of q = exp(Pi i t).

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