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A007331
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Fourier coefficients of E_{\infty,4}.
(Formerly M4503)
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12
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| E_{\infty,4} is the unique normalized weight-4 modular form for \Gamma_0(2) with simple zeros at i*\infty. Since this has level 2, it is not a cusp form, in contrast to A002408.
Number of representations of n-1 as sum of 8 triangular numbers.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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REFERENCES
| B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139, Ex (ii).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 187.
Masao Koike, Modular forms on non-compact arithmetic triangle groups, preprint.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1001
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
H. Rosengren, Sums of triangular numbers from the Frobenius determinant
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FORMULA
| G.f.: q * Product (1+q^(2*k-1))^8*(1+q^(4*k))^8, k=1..inf.
a(n) = sum_{0<d|n, n/d odd} d^3.
G.f.: Sum_{n>0} n^3*x^n/(1-x^(2*n)). - Vladeta Jovovic (vladeta(AT)eunet.rs), 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 g() is 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 (Harris.Mitchell(AT)mgh.harvard.edu), 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).
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EXAMPLE
| 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
| q*product( (1+q^(2*k-1))^8*(1+q^(4*k))^8, k=1..75);
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MATHEMATICA
| Prepend[Table[Plus @@ (Select[Divisors[k + 1], OddQ[(k + 1)/#] &]^3), {k, 0, 39}], 0] (*Ant King, Dec 04 2010*)
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PROG
| (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 */
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CROSSREFS
| Cf. A002408, A004017, A035016, A045825, A076577, A096960.
Sequence in context: A173681 A045850 A033580 * A002408 A101127 A007259
Adjacent sequences: A007328 A007329 A007330 * A007332 A007333 A007334
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KEYWORD
| easy,nice,nonn,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein (mira(AT)math.berkeley.edu)
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EXTENSIONS
| Additional comments from Barry Brent (barryb(AT)primenet.com)
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