A286131 - OEIS

%I #19 Mar 12 2021 22:24:48

%S 0,0,0,0,1,-1,-1,0,0,1,0,1,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,1,0,0,0,1,0,

%T 0,0,-1,1,1,0,0,-3,1,0,0,0,-2,0,-1,1,1,1,0,0,0,-1,1,1,-1,0,1,0,-1,1,0,

%U 0,-1,0,1,0,-1,1,-1,-2,-1,0,1,1,4,-1,-1,1,0,0

%N Expansion of q^(-1/2) * eta(q) * eta(q^30) * eta(q^35) * eta(q^42) in powers of q.

%C Euler transform of a period 210 sequence. - _Michael Somos_, Nov 14 2019

%H Seiichi Manyama, <a href="/A286131/b286131.txt">Table of n, a(n) for n = 0..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^4 * Prod_{k>0} (1 - x^k) * (1 - x^(30 * k)) * (1 - x^(35 * k)) * (1 - x^(42 * k)).

%e G.f. = x^4 - x^5 - x^6 + x^9 + x^11 - x^16 - x^19 + x^26 + x^30 - x^34 + ... - _Michael Somos_, Nov 14 2019

%p seq(coeff(series(x^4*mul((1-x^k)*(1-x^(30*k))*(1-x^(35*k))*(1-x^(42*k)),k=1..n), x,n+1),x,n),n=0..150); # _Muniru A Asiru_, Jul 29 2018

%t eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/2) *eta[q]*eta[q^30]*eta[q^35]*eta[q^42], {q, 0, 50}], q] (* _G. C. Greubel_, Jul 29 2018 *)

%o (PARI) q='q+O('q^50); A = eta(q)*eta(q^30)*eta(q^35)*eta(q^42); concat(vector(4), Vec(A)) \\ _G. C. Greubel_, Jul 29 2018

%o (PARI) {a(n) = n-=4; if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^30 + A) * eta(x^35 + A) * eta(x^42 + A), n))}; /* _Michael Somos_, Nov 14 2019 */

%Y Cf. A286135.

%K sign

%O 0,40

%A _Seiichi Manyama_, May 03 2017