A122776 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.

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, -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, 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

Offset

1,4

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Michael Somos, A Remarkable eta-product Identity

Formula

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)).

a(n) = -A030184(2*n).

a(2n-1) = A030184(2n-1), a(2n) = A030184(2n) + 2 * A030184(n). - Seiichi Manyama, May 03 2017

a(n) = A030218(n) + A286137(n). - Seiichi Manyama, May 03 2017

Prog

(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))};
(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))};
(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)))};

Crossrefs

Cf. A030184, A030218, A286137.

Sequence in context: A333351 A193140 A319993 * A261609 A228488 A338638

Adjacent sequences: A122773 A122774 A122775 * A122777 A122778 A122779

Keyword

sign,mult

Author

Michael Somos, Sep 10 2006

Status

approved