Search: id:A000735
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%I A000735 M4841 N2069
%S A000735 1,12,54,88,99,540,418,648,594,836,1056,4104,209,4104,594,4256,6480,
%T A000735 4752,298,5016,17226,12100,5346,1296,9063,7128,19494,29160,10032,7668,
%U A000735 34738,8712,22572,21812,49248,46872,67562,2508,47520,76912,25191,67716
%V A000735 1,-12,54,-88,-99,540,-418,-648,594,836,1056,-4104,-209,4104,-594,4256,
               -6480,-4752,
%W A000735 -298,5016,17226,-12100,-5346,-1296,-9063,-7128,19494,29160,-10032,-7668,
               -34738,8712,
%X A000735 -22572,21812,49248,-46872,67562,2508,-47520,-76912,-25191,67716
%N A000735 Expansion of Product (1-x^k)^(12).
%C A000735 Grosswald uses b_n where b_{2n+1} = a(n).
%C A000735 A000145(n)=A029751(n)+16*a(n). - Michael Somos Sep 21 2005
%D A000735 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000735 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000735 A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J., 
               8 (1967), 14-32.
%D A000735 M. Boylan, Exceptional congruences for the coefficients of certain eta-product 
               newforms, J. Number Theory 98 (2003), no. 2, 377-389.
%D A000735 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.
%D A000735 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).
%D A000735 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, 
               NY, 1985, p. 121.
%H A000735 T. D. Noe, <a href="b000735.txt">Table of n, a(n) for n=0..1000</a>
%H A000735 <a href="Sindx_Pro.html#1mxtok">Index entries for expansions of Product_{k 
               >= 1} (1-x^k)^m</a>
%F A000735 Expansion of q^(-1/2) eta(q)^12 in powers of q.
%F A000735 Euler transform of period 1 sequence [ -12, ...]. - Michael Somos Sep 
               21 2005
%F A000735 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
%F A000735 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
%F A000735 G.f.: (Product_{k>0} (1-x^k))^12.
%e A000735 B(x) = x - 12*x^3 + 54*x^5 - 88*x^7 - 99*x^9 + 540*x^11 - 418*x^13 - 
               ...
%p A000735 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]
%o A000735 (PARI) a(n)=if(n<0, 0, polcoeff(eta(x+x*O(x^n))^12,n)) /* Michael Somos 
               Sep 21 2005 */
%Y A000735 Sequence in context: A060171 A133078 A034436 this_sequence A022704 A060785 
               A059986
%Y A000735 Adjacent sequences: A000732 A000733 A000734 this_sequence A000736 A000737 
               A000738
%K A000735 sign,easy,nice
%O A000735 0,2
%A A000735 N. J. A. Sloane (njas(AT)research.att.com).

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