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%I M4503 #148 Oct 21 2022 10:07:24
%S 0,1,8,28,64,126,224,344,512,757,1008,1332,1792,2198,2752,3528,4096,
%T 4914,6056,6860,8064,9632,10656,12168,14336,15751,17584,20440,22016,
%U 24390,28224,29792,32768,37296,39312,43344,48448,50654,54880,61544,64512
%N Fourier coefficients of E_{infinity,4}.
%C 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.
%C 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.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C 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
%D B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139, Ex (ii).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A007331/b007331.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1001 from T. D. Noe)
%H B. Brent, <a href="http://projecteuclid.org/euclid.em/1047674207">Quadratic Minima and Modular Forms</a>, Experimental Mathematics, v.7 no.3, 257-274.
%H H. H. Chan and C. Krattenthaler, <a href="https://arxiv.org/abs/math/0407061">Recent progress in the study of representations of integers as sums of squares</a>, arXiv:math/0407061 [math.NT], 2004.
%H J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1007/978-1-4757-2016-7">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, p. 187.
%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
%H Masao Koike, <a href="https://oeis.org/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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.]
%H K. Ono, S. Robins and P. T. Wahl, <a href="https://www.researchgate.net/publication/227141445_On_the_representation_of_integer_as_sum_of_triangular_numbers">On the representation of integers as sums of triangular numbers</a>, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Theorem 5.
%H H. Rosengren, <a href="https://arxiv.org/abs/math/0504272">Sums of triangular numbers from the Frobenius determinant</a>, arXiv:math/0504272 [math.NT], 2005.
%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>.
%F G.f.: q * Product_{k>=1} (1-q^k)^8 * (1+q^k)^16. - corrected by _Vaclav Kotesovec_, Oct 14 2015
%F a(n) = Sum_{0<d|n, n/d odd} d^3. [Glaisher]
%F G.f.: Sum_{n>0} n^3*x^n/(1-x^(2*n)). - _Vladeta Jovovic_, Oct 24 2002
%F Expansion of Jacobi theta constant theta_2(q)^8 / 256 in powers of q.
%F Expansion of eta(q^2)^16 / eta(q)^8 in powers of q. - _Michael Somos_, May 31 2005
%F Expansion of x * psi(x)^8 in powers of x where psi() is a Ramanujan theta function. - _Michael Somos_, Jan 15 2012
%F Expansion of (Q(x) - Q(x^2)) / 240 in powers of x where Q() is a Ramanujan Lambert series. - _Michael Somos_, Jan 15 2012
%F Expansion of E_{gamma,2}^2 * E_{0,4} in powers of q.
%F Euler transform of period 2 sequence [8, -8, ...]. - _Michael Somos_, May 31 2005
%F 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
%F 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
%F Multiplicative with a(2^e) = 2^(3e), a(p^e) = (p^(3(e+1))-1)/(p^3-1). - _Mitch Harris_, Jun 13 2005
%F Dirichlet convolution of A154955 by A001158. Dirichlet g.f. zeta(s)*zeta(s-3)*(1-1/2^s). - _R. J. Mathar_, Mar 31 2011
%F A002408(n) = -(-1)^n * a(n).
%F Convolution square of A008438. - _Michael Somos_, Jun 15 2014
%F 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
%F Sum_{k=1..n} a(k) ~ c * n^4, where c = Pi^4/384 = 0.253669... (A222072). - _Amiram Eldar_, Oct 19 2022
%e G.f. = q + 8*q^2 + 28*q^3 + 64*q^4 + 126*q^5 + 224*q^6 + 344*q^7 + 512*q^8 + ...
%p 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
%t Prepend[Table[Plus @@ (Select[Divisors[k + 1], OddQ[(k + 1)/#] &]^3), {k, 0, 39}], 0] (* _Ant King_, Dec 04 2010 *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^8 / 256, {q, 0, n}]; (* _Michael Somos_, Jun 04 2013 *)
%t a[ n_] := If[ n < 1, 0, Sum[ d^3 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* _Michael Somos_, Jun 04 2013 *)
%t f[n_] := Total[(2n/Select[ Divisors[ 2n], Mod[#, 4] == 2 &])^3]; Flatten[{0, Array[f, 40] }] (* _Robert G. Wilson v_, Mar 26 2015 *)
%t 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 *)
%t 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 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^3))}; /* _Michael Somos_, May 31 2005 */
%o (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 */
%o (PARI) a(n)=my(e=valuation(n,2)); 8^e * sigma(n/2^e, 3) \\ _Charles R Greathouse IV_, Sep 09 2014
%o (Sage) ModularForms( Gamma0(2), 4, prec=33).1; # _Michael Somos_, Jun 04 2013
%o (Magma) Basis( ModularForms( Gamma0(2), 4), 10) [2]; /* _Michael Somos_, May 27 2014 */
%o (Python)
%o from sympy import divisors
%o def a(n):
%o return 0 if n == 0 else sum(((n//d)%2)*d**3 for d in divisors(n))
%o print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jun 24 2017
%Y Cf. A002408, A004017, A035016, A045825, A076577, A096960, A222072.
%Y 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.
%K easy,nice,nonn,mult
%O 0,3
%A _N. J. A. Sloane_, _Mira Bernstein_
%E Additional comments from Barry Brent (barryb(AT)primenet.com)
%E Wrong Maple program replaced by _Vaclav Kotesovec_, Oct 14 2015
%E a(0)=0 prepended by _Vaclav Kotesovec_, Oct 14 2015
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