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A007331
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Fourier coefficients of E_{infinity,4}.
(Formerly M4503)
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38
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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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Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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
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FORMULA
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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<d|n, n/d odd} d^3. [Glaisher]
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
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EXAMPLE
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G.f. = q + 8*q^2 + 28*q^3 + 64*q^4 + 126*q^5 + 224*q^6 + 344*q^7 + 512*q^8 + ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^3))}; /* Michael Somos, May 31 2005 */
(PARI) {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 */
(PARI) a(n)=my(e=valuation(n, 2)); 8^e * sigma(n/2^e, 3) \\ Charles R Greathouse IV, Sep 09 2014
(Sage) ModularForms( Gamma0(2), 4, prec=33).1; # Michael Somos, Jun 04 2013
(MAGMA) Basis( ModularForms( Gamma0(2), 4), 10) [2]; /* Michael Somos, May 27 2014 */
(Python)
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
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CROSSREFS
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Cf. A002408, A004017, A035016, A045825, A076577, A096960.
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.
Sequence in context: A299289 A212515 A342251 * A002408 A340964 A101127
Adjacent sequences: A007328 A007329 A007330 * A007332 A007333 A007334
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KEYWORD
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easy,nice,nonn,mult
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AUTHOR
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N. J. A. Sloane, Mira Bernstein
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EXTENSIONS
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Additional comments from Barry Brent (barryb(AT)primenet.com)
Wrong Maple program replaced by Vaclav Kotesovec, Oct 14 2015
Prepended a(0)=0. - Vaclav Kotesovec, Oct 14 2015
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STATUS
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approved
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