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A000727
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Expansion of Product_{k >= 1} (1-x^k)^4.
(Formerly M3204 N1296)
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3
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1, -4, 2, 8, -5, -4, -10, 8, 9, 0, 14, -16, -10, -4, 0, -8, 14, 20, 2, 0, -11, 20, -32, -16, 0, -4, 14, 8, -9, 20, 26, 0, 2, -28, 0, -16, 16, -28, -22, 0, 14, 16, 0, 40, 0, -28, 26, 32, -17, 0, -32, -16, -22, 0, -10, 32, -34, -8, 14, 0, 45, -4, 38, 8, 0, 0, -34, -8, 38, 0, -22, -56, 2, -28, 0, 0, -10, 20, 64, -40, -20, 44
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Newman, Morris; A table of the coefficients of the powers of $\eta(\tau)$. Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 415. Exer. 47.2.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Robert M. Ziff, "On Cardy's formula for the critical crossing probability in 2d percolation," J. Phys. A. 28, 1249-1255 (1995).
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LINKS
| Alois P. Heinz, Table of n, a(n) for n = 0..10000
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
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FORMULA
| Euler transform of period 1 sequence [ -4, -4, ...]. - Michael Somos Apr 2 2005
Given g.f. A(x), then B(x)=x*A(x^3)^2 satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=wu^2-v^3+16uw^2. - Michael Somos Apr 2 2005
a(n)=b(6n+1) and b(n) is multiplicative with b(2^e)=b(3^e)=0^e, b(p^e)=b(p)b(p^(e-1))-p*b(p^(e-2)), b(p)=0 if p == 5 (mod 6), b(p)=2x where p=x^2+3y^2 == 1 (mod 6) and x == 1 (mod 3). - Michael Somos Aug 23 2006
Coefficients of L-series for elliptic curve "36a1": y^2= x^3 +1 . - Michael Somos Jul 1 2004
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MAPLE
| with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr (n-> -4): seq (a(n), n=0..81); # Alois P. Heinz, Sep 08 2008
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PROG
| (PARI) {a(n)= local(A, p, e, x, y, a0, a1); if(n<0, 0, n=6*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p<5, 0, if(p%6==5, if(e%2, 0, (-1)^(e/2)*p^(e/2)), for(y=1, sqrtint(p\3), if(issquare(p-3*y^2, &x), break)); a0=1; if(x%3!=1, x=-x); a1=x=2*x; for(i=2, e, y=x*a1-p*a0; a0=a1; a1=y); a1)))))} /* Michael Somos Aug 23 2006 */
(PARI) {a(n)= if(n<0, 0, polcoeff(eta(x +x*O(x^n))^4, n))}
(PARI) {a(n)= if(n<0, 0, n= 6*n +1; ellak( ellinit( [0, 0, 0, 0, 1]), n))} /* Michael Somos Jul 1 2004 */
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CROSSREFS
| Sequence in context: A194054 A191536 A187076 * A030181 A021879 A020806
Adjacent sequences: A000724 A000725 A000726 * A000728 A000729 A000730
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KEYWORD
| sign
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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