A261278 - OEIS

%I #8 Sep 08 2022 08:46:13

%S 1,4,0,-8,0,0,20,-32,0,0,0,0,-70,80,0,64,0,0,56,0,0,0,0,0,-125,-280,0,

%T -160,0,0,308,256,0,0,0,0,110,224,0,0,0,0,-520,0,0,0,0,0,57,-500,0,

%U 560,0,0,0,-640,0,0,0,0,182,1232,0,-512,0,0,-880,0,0,0,0

%N Expansion of eta(q^3)^8 + 4 * eta(q^6)^8 in powers of q.

%H G. C. Greubel, <a href="/A261278/b261278.txt">Table of n, a(n) for n = 1..2500</a>

%F a(n) is multiplicative with a(2^(2*k)) = (-8)^k, a(2^(2*k+1)) = 4 * (-8)^k, a(3^e) = 0^e, a(p^(2*k)) = (-p)^(3^k) and a(p^(2*k+1)) = 0 if p == 5 (mod 6), a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)) if p == 1 (mod 6).

%F a(3*n) = a(6*n + 5) = 0. a(3*n + 1) = A000731(n). a(4*n) = -8 * a(n). a(6*n + 1) = A153728(n).

%F Convolution square of A261277.

%e G.f. = x + 4*x^2 - 8*x^4 + 20*x^7 - 32*x^8 - 70*x^13 + 80*x^14 + ...

%t a[ n_] := SeriesCoefficient[ x QPochhammer[ x^3]^8 + 4 x^2 QPochhammer[ x^6]^8, {x, 0, n}];

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^8 + 4 * x * eta(x^6 + A)^8, n))};

%o (Magma) A := Basis( CuspForms( Gamma0(18), 4), 72); A[1] + 4*A[2] - 8*A[4];

%o (Sage) A = CuspForms( Gamma0(18), 4, prec=20).basis(); A[0] + 4*A[1] - 8*A[3];

%Y Cf. A000731, A153728, A261277.

%K sign,mult

%O 1,2

%A _Michael Somos_, Aug 14 2015