A143752 Expansion of eta(q^3) * eta(q^4) * eta(q^5) * eta(q^60) / (eta(q) * eta(q^12) * eta(q^15) * eta(q^20)) in powers of q.

1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 10, 11, 14, 17, 20, 23, 27, 31, 36, 41, 48, 55, 63, 72, 82, 94, 106, 122, 137, 156, 175, 197, 222, 249, 280, 314, 352, 393, 439, 490, 546, 608, 676, 751, 834, 923, 1024, 1133, 1253, 1384, 1528, 1686, 1857, 2045, 2250, 2474, 2718

Offset

1,3

Links

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

Michael Somos, A Remarkable eta-product Identity

Formula

Expansion of F(q) * F(q^2) in powers of q^3 where F(q) is the g.f. for A103263.

Euler transform of a period 60 sequence.

G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143751.

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u^2 + v^2) * (1 + u + v) * (u + v + u*v) - u*v * (1+ 2*u + 2*v + u*v)^2.

G.f.: x * Product_{k>0} P(30, x^k) * P(60, x^k) where P(n, x) is the n-th cyclotomic polynomial.

a(2*n) = A123630(n). Convolution inverse of A143751.

G.f.: -1 + Product_{k>0} (1 + x^k) * (1 + x^(15*k)) / ((1 + x^(6*k)) * (1 + x^(10*k))). - Seiichi Manyama, May 04 2017

a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018

Example

G.f. = q + q^2 + 2*q^3 + 2*q^4 + 3*q^5 + 3*q^6 + 4*q^7 + 5*q^8 + 6*q^9 + 7*q^10 + ...

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^60 + A) / (eta(x + A) * eta(x^12 + A) * eta(x^15 + A) * eta(x^20 + A)), n))};

Crossrefs

Cf. A123630, A143751, A145933.

Sequence in context: A029023 A140952 A096911 * A145933 A300788 A120171

Adjacent sequences: A143749 A143750 A143751 * A143753 A143754 A143755

Keyword

nonn

Author

Michael Somos, Aug 31 2008

Status

approved