A002408 -id:A002408 - cp's OEIS frontend

A007331 Fourier coefficients of E_{infinity,4}.

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Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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

A002434 Theta series of Borcherds' 27-dimensional unimodular lattice T_27.

Original entry on oeis.org

1, 0, 0, 1640, 119574, 1497600, 16733184, 108081792, 588805308, 2544826368, 9516533760, 31328289720, 92876121704, 252846217728, 638250227712, 1511780699520, 3387237774102, 7228330481664

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
R. Bacher and B. B. Venkov, Réseaux entiers unimodulaires sans racine en dimension 27 et 28, in Réseaux euclidiens, designs sphériques et formes modulaires, pp. 212-267, Enseignement Math., Geneva, 2001.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Preface to 3rd ed.

th3^27-54*th3^19*delta8+216*th3^11*delta8^2-1024*th3^3*delta8^3 # (th3= A000122, delta8= A002408).

terms = 18; QP = QPochhammer; th3 = EllipticTheta[3, 0, q]; delta8 = q*(QP[q]*(QP[q^4]/QP[q^2]))^8; s = th3^27 - 54*th3^19*delta8 + 216*th3^11*delta8^2 - 1024*th3^3*delta8^3 + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 06 2017 *)

A002490 Theta series of 27-dimensional unimodular lattice with root system A_1 and a parity vector of norm 3.

Original entry on oeis.org

1, 0, 2, 1652, 119550, 1497328, 16733124, 108084480, 588808172, 2544811584, 9516509988, 31328336284, 92876219432, 252846150384, 638250041288, 1511780647680, 3387237676950, 7228330859840, 14769380958438, 29023337216604, 55108233751768, 101433859301088

Offset: 0

N. J. A. Sloane

nonn

R. Bacher and B. B. Venkov, Réseaux entiers unimodulaires sans racine en dimension 27 et 28, in Réseaux euclidiens, designs sphériques et formes modulaires, pp. 212-267, Enseignement Math., Geneva, 2001.

R. Bacher and B. B. Venkov, Réseaux entiers unimodulaires sans racine en dimension 27 et 28

Cf. A000122, A002408, A002434.

G.f.: theta3^27 - 54*theta3^19*delta8 + 218*theta3^11*delta8^2 - 1024*theta3^3*delta8^3 where theta3 = A000122 and delta8 = A002408. - Sean A. Irvine, Feb 28 2020

More terms from Sean A. Irvine, Feb 28 2020

A002482 Theta series of Borcherds' 27-dimensional unimodular lattice U_27.

Original entry on oeis.org

1, 0, 0, 2664, 101142, 1645056, 16045056, 110146176, 584713404, 2549741568, 9515943936, 31314087864, 92917622376, 252775586304, 638328674304, 1511740886400, 3387163161366, 7228598851584

Offset: 0

N. J. A. Sloane

nonn

R. Bacher and B. B. Venkov, Réseaux entiers unimodulaires sans racine en dimension 27 et 28, in Réseaux euclidiens, designs sphériques et formes modulaires, pp. 212-267, Enseignement Math., Geneva, 2001.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Preface to 3rd ed. [alternative link]
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.

th3^27-54*th3^19*delta8+216*th3^11*delta8^2 (th3 = A000122, delta8 = A002408).

terms = 18; QP = QPochhammer; th3 = EllipticTheta[3, 0, q]; delta8 = q*(QP[q]*(QP[q^4]/QP[q^2]))^8; s = th3^27 - 54*th3^19*delta8 + 216*th3^11*delta8^2 + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 06 2017 *)

A002495 Theta series of 27-dimensional unimodular lattice with root system A_1 with no parity vector of norm 3.

Original entry on oeis.org

1, 0, 2, 2676, 101118, 1644784, 16044996, 110148864, 584716268, 2549726784, 9515920164, 31314134428, 92917720104, 252775518960, 638328487880, 1511740834560, 3387163064214, 7228599229760, 14768913424614, 29023937463900, 55107648867544, 101434033397472

Offset: 0

N. J. A. Sloane

nonn

R. Bacher and B. B. Venkov, Réseaux entiers unimodulaires sans racine en dimension 27 et 28, in Réseaux euclidiens, designs sphériques et formes modulaires, pp. 212-267, Enseignement Math., Geneva, 2001.

R. Bacher and B. B. Venkov, Réseaux entiers unimodulaires sans racine en dimension 27 et 28

Cf. A000122, A002408, A002434, A002490.

G.f.: theta3^27 - 54*theta3^19*delta8 + 218*theta3^11*delta8^2 where theta3 = A000122 and delta8 = A002408. - Sean A. Irvine, Feb 28 2020

More terms from Sean A. Irvine, Feb 28 2020

A034998 Expansion of Product (1+q^(2k-1))^(-8)*(1+q^(4k))^(-8), k=1..inf.

Original entry on oeis.org

1, -8, 36, -128, 402, -1152, 3064, -7680, 18351, -42112, 93300, -200448, 419150, -855552, 1708632, -3345408, 6432867, -12166272, 22659976, -41609856, 75404754, -134973184, 238825344, -418023936, 724242492

Offset: 0

N. J. A. Sloane

sign

Reciprocal of A007331. Cf. A002408.

A066234 Theta series of GH39, an extremal unimodular 39-dimensional lattice.

Original entry on oeis.org

1, 0, 0, 0, 23790, 1525056, 44420480, 768892800, 9093033300, 80361232640, 564369691392, 3290997166080, 16459430933400, 72362333294400, 285055580709120, 1021529151913088, 3371180795232510, 10348222578720000

Offset: 0

Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Dec 19 2001

nonn

T. Gulliver and M. Harada, An optimal unimodular lattice in dimension 39, J. Combinat. Theory A 88, 158-161 (1999).

Cf. A000122 (theta_3), A002408 (Del8).

theta_3^39 - 78*theta_3^31*Del8 + 1248*theta_3^23*Del8^2 - 1976*theta_3^15*Del8^3.

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cp's OEIS Frontend

A302201 E.g.f.: exp (e.g.f. for the "cusp form" A002408).

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A007331 Fourier coefficients of E_{infinity,4}.

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A002434 Theta series of Borcherds' 27-dimensional unimodular lattice T_27.

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A002490 Theta series of 27-dimensional unimodular lattice with root system A_1 and a parity vector of norm 3.

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A002482 Theta series of Borcherds' 27-dimensional unimodular lattice U_27.

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A002495 Theta series of 27-dimensional unimodular lattice with root system A_1 with no parity vector of norm 3.

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A034998 Expansion of Product (1+q^(2k-1))^(-8)*(1+q^(4k))^(-8), k=1..inf.

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A066234 Theta series of GH39, an extremal unimodular 39-dimensional lattice.

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