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Search: id:A000735
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| A000735 |
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Expansion of Product (1-x^k)^(12).
(Formerly M4841 N2069)
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+0
3
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| 1, -12, 54, -88, -99, 540, -418, -648, 594, 836, 1056, -4104, -209, 4104, -594, 4256, -6480, -4752, -298, 5016, 17226, -12100, -5346, -1296, -9063, -7128, 19494, 29160, -10032, -7668, -34738, 8712, -22572, 21812, 49248, -46872, 67562, 2508, -47520, -76912, -25191, 67716
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Grosswald uses b_n where b_{2n+1} = a(n).
A000145(n)=A029751(n)+16*a(n). - Michael Somos Sep 21 2005
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J., 8 (1967), 14-32.
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389.
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. W. L. Glaisher, On the representation of a number as sum of 2,4,6,8... squares, Quart. J. Math. 38 (1907), 1-62 (see p. 56).
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000
Index entries for expansions of Product_{k >= 1} (1-x^k)^m
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FORMULA
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Expansion of q^(-1/2) eta(q)^12 in powers of q.
Euler transform of period 1 sequence [ -12, ...]. - Michael Somos Sep 21 2005
Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)=u^4*w^2+48*(u*v*w)^2+4906*u^2*w^4-u^6 . - Michael Somos Sep 21 2005
a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = b(p)*b(p^(e-1)) -p^5*b(p^(e-2)) . - Michael Somos Mar 08 2006
G.f.: (Product_{k>0} (1-x^k))^12.
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EXAMPLE
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B(x) = x - 12*x^3 + 54*x^5 - 88*x^7 - 99*x^9 + 540*x^11 - 418*x^13 - ...
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MAPLE
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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-> -12): seq (a(n), n=0..41); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 08 2008]
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PROGRAM
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(PARI) a(n)=if(n<0, 0, polcoeff(eta(x+x*O(x^n))^12, n)) /* Michael Somos Sep 21 2005 */
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CROSSREFS
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Sequence in context: A060171 A133078 A034436 this_sequence A022704 A060785 A059986
Adjacent sequences: A000732 A000733 A000734 this_sequence A000736 A000737 A000738
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KEYWORD
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sign,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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