cp's OEIS Frontend

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.

%I A143752 #16 Mar 12 2021 22:24:45
%S A143752 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,
%T A143752 94,106,122,137,156,175,197,222,249,280,314,352,393,439,490,546,608,
%U A143752 676,751,834,923,1024,1133,1253,1384,1528,1686,1857,2045,2250,2474,2718
%N 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.
%H A143752 Seiichi Manyama, <a href="/A143752/b143752.txt">Table of n, a(n) for n = 1..10000</a>
%H A143752 Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/retaprod.html">A Remarkable eta-product Identity</a>
%F A143752 Expansion of F(q) * F(q^2) in powers of q^3 where F(q) is the g.f. for A103263.
%F A143752 Euler transform of a period 60 sequence.
%F A143752 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.
%F A143752 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.
%F A143752 G.f.: x * Product_{k>0} P(30, x^k) * P(60, x^k) where P(n, x) is the n-th cyclotomic polynomial.
%F A143752 a(2*n) = A123630(n). Convolution inverse of A143751.
%F A143752 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
%F A143752 a(n) ~ exp(2*Pi*sqrt(n/15)) / (2 * 15^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 03 2018
%e A143752 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 + ...
%o A143752 (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))};
%Y A143752 Cf. A123630, A143751, A145933.
%K A143752 nonn
%O A143752 1,3
%A A143752 _Michael Somos_, Aug 31 2008