A298107 - OEIS

%I #16 Jul 26 2018 07:52:58

%S 1,2,5,10,18,30,51,80,124,190,281,410,592,840,1178,1640,2253,3070,

%T 4154,5570,7422,9830,12932,16920,22028,28520,36761,47180,60280,76720,

%U 97278,122880,154693,194110,242776,302740,376424,466710,577114,711800,875707,1074790

%N Expansion of (eta(q^4) * eta(q^5) / (eta(q) * eta(q^20)))^2 in powers of q.

%H G. C. Greubel, <a href="/A298107/b298107.txt">Table of n, a(n) for n = -1..1500</a>

%F Euler transform of period 20 sequence [2, 2, 2, 0, 0, 2, 2, 0, 2, 0, 2, 0, 2, 2, 0, 0, 2, 2, 2, 0, ...].

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = f(t) where q = exp(2 Pi i t).

%F G.f.: 1/x * Product_{k>0} ((1 - x^(4*k)) * (1 - x^(5*k)))^2 / ((1 - x^k) * (1 - x^(20*k)))^2.

%F a(n) = A058555(n) unless n=0. Convolution square of A058664.

%F a(n) ~ exp(2*Pi*sqrt(n/5)) / (2*5^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Mar 21 2018

%e G.f. = q^-1 + 2 + 5*q + 10*q^2 + 18*q^3 + 30*q^4 + 51*q^5 + 80*q^6 + ...

%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^4] QPochhammer[ q^5])^2 / (QPochhammer[ q] QPochhammer[ q^20])^2, {q, 0 , n}];

%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^4 + A) * eta(x^5 + A) / (eta(x + A) * eta(x^20 + A)))^2, n))};

%Y Cf. A058555, A058664.

%K nonn

%O -1,2

%A _Michael Somos_, Jan 12 2018