|
%I #19 May 31 2022 16:28:15
%S 1,1,-1,-3,1,-1,0,1,1,1,-4,3,-2,0,-1,5,2,1,4,-3,0,-4,0,-1,1,-2,-1,0,
%T -2,-1,0,-7,4,2,0,-3,-10,4,2,1,10,0,4,12,1,0,8,-5,-7,1,-2,6,-10,-1,-4,
%U 0,-4,-2,-4,3,-2,0,0,-3,-2,4,12,-6,0,0,-8,1,10,-10,-1,-12,0,2,0,5,1,10,12,0,2,4,2,-4,-6,1,0,0,0,8,4,7,2,-7,-4
%N Expansion of eta(q^3)eta(q^5)eta(q^6)eta(q^10) + eta(q)eta(q^2)eta(q^15)eta(q^30) in powers of q.
%H Seiichi Manyama, <a href="/A122776/b122776.txt">Table of n, a(n) for n = 1..10000</a>
%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/retaprod.html">A Remarkable eta-product Identity</a>
%F G.f.: x Product_{k>0} (1-x^(3k))(1-x^(5k))(1-x^(6k))(1-x^(10k)) + x^2 Product_{k>0} (1-x^k)(1-x^(2k))(1-x^(15k))(1-x^(30k)).
%F a(n) = -A030184(2*n).
%F a(2n-1) = A030184(2n-1), a(2n) = A030184(2n) + 2 * A030184(n). - _Seiichi Manyama_, May 03 2017
%F a(n) = A030218(n) + A286137(n). - _Seiichi Manyama_, May 03 2017
%o (PARI) {a(n) = my(A); if( n<1, 0, n*=2; n--; A = x * O(x^n); polcoeff( -eta(x + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^15 + A), n))};
%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^5 + A) * eta(x^6 + A) * eta(x^10 + A) + eta(x+A) * eta(x^2 + A) * eta(x^15 + A) * eta(x^30 + A)*x, n))};
%o (PARI) {a(n) = my(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==3, (-1)^e, p==5, 1, a0=1; y=if(p==2, a1=1; -1, a1=-sum(x=0, p-1, kronecker(4*x^3+5*x^2+2*x+1,p))); for(i=2, e, x=y*a1-p*a0; a0=a1; a1=x); a1)))};
%Y Cf. A030184, A030218, A286137.
%K sign,mult
%O 1,4
%A _Michael Somos_, Sep 10 2006
|
