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A002408
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Expansion of 8-dimensional cusp form.
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5
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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, 68922
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).
"For Gamma, it is known that any modular form is a weighted homogeneous polynomial in Theta_Z, which has weight 1/2 and the modular form delta_8(t) := e^{pi i tau} prod_{m=1..infty} ((1-e^{ pi i m tau}) (1+e^{2 pi i m tau}))^8 = e^{ pi i tau} - 8 e^{2 pi i m tau} +28 e^{3 pi i m tau} -64 e^{4 pi i m tau} +126 e^{5 pi i m tau} ... of weight 4." [Elkies, p. 1242]
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REFERENCES
| J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 187.
N. D. Elkies. Lattices, Linear Codes and Invariants, Part I. Amer. Math. Soc., 47 (No. 10, Nov. 2000), 1238-1245, see p. 1242.
F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg 1994 p 133.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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FORMULA
| Expansion of (eta(q)eta(q^4)/eta(q^2))^8 in powers of q. - Michael Somos, Jul 16 2004
Euler transform of period 4 sequence [ -8,0,-8,-8,...]. - Michael Somos, Jul 16 2004
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^4)) where f(u, v, w)= +u^4*w*v +16*u^3*w*v^2 +16*u^2*w^2*v^2 +256*u^3*w^3 +256*u^3*w^2*v +4096*u^2*w^3*v +4096*u*w^4*v +4096*u*w^3*v^2 -u^2*v^4 -16*u^2*w*v^3 -256*u*w^2*v^3 -256*w^2*v^4 . - Michael Somos May 31 2005
G.f. A(x) satisfies 0=f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6)=u1^4*u6^4 +u1^3*u2*u3^3*u6 +2*u1*u2^3*u3*u6^3 -u2^4*u3^4.
Expansion of q * psi(-q)^8 in powers of q where psi() is a Ramanujan theta function. - Michael Somos Mar 20 2008
a(n) is multiplicative with a(2^e) = -8^e if e>0, a(p^e) = ((p^3)^(e+1) - 1) / (p^3 - 1). - Michael Somos Mar 20 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 16 (t/i)^4 f(t) where q = exp(2 pi i t).
G.f.: x * (Product_{k>0} (1 - x^(2*k-1)) * (1 - x^(4*k)))^8.
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EXAMPLE
| q - 8*q^2 + 28*q^3 - 64*q^4 + 126*q^5 - 224*q^6 + 344*q^7 ...
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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
| a[0] = 0; a[n_] := -(-1)^n*Sum[ Mod[n/d, 2]*d^3, {d, Divisors[n]}]; Table[a[n], {n, 0, 41}] (* From Jean-François Alcover, Jan 27 2012, after Michael Somos *)
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PROG
| (PARI) a(n)=local(A); if(n<1, 0, n--; A=x^n*O(x); polcoeff((eta(x+A)/eta(x^2+A)*eta(x^4+A))^8, n)) /* Michael Somos, Jul 16 2004 */
(PARI) a(n)=local(A); if(n<1, 0, n--; A=x^n*O(x); polcoeff((prod(k=1, n, (1-(k%4==0)*x^k)*(1-(k%2==1)*x^k), 1+A))^8, n)) /* Michael Somos, Jul 16 2004 */
(PARI) a(n)=if(n<1, 0, -(-1)^n*sumdiv(n, d, (n/d%2)*d^3)) /* Michael Somos May 31 2005 */
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CROSSREFS
| a(n)=-(-1)^n*A007331(n).
Sequence in context: A045850 A033580 A007331 * A101127 A007259 A134747
Adjacent sequences: A002405 A002406 A002407 * A002409 A002410 A002411
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KEYWORD
| sign,nice,easy,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com) and Mira Bernstein (mira(AT)math.berkeley.edu)
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