A232506 - OEIS

%I #18 Sep 08 2022 08:46:06

%S 1,-2,-1,2,1,2,-2,0,-2,-2,1,0,0,2,3,-2,2,0,0,-2,-2,0,0,-4,3,2,-2,0,-6,

%T 6,1,6,4,0,-2,-2,-2,-6,2,-4,-4,0,4,4,1,-2,3,4,3,-6,-3,4,-1,-4,-2,4,-3,

%U 4,4,-8,-3,4,-2,6,2,2,-2,-2,4,2,-4,-4,2,-2,2,-8

%N Expansion of (eta(q) * eta(q^23))^2 in powers of q.

%H G. C. Greubel, <a href="/A232506/b232506.txt">Table of n, a(n) for n = 2..1000</a>

%H John F. R. Duncan, Michael J. Griffin and Ken Ono, <a href="http://arxiv.org/abs/1503.01472">Proof of the Umbral Moonshine Conjecture</a>, arXiv:1503.01472, 2015. See Eq. (B.31).

%F Expansion of a level 2 Gamma0(23) cusp form in powers of q with a(1) = 0, a(2) = 1.

%F Euler transform of period 23 sequence [ -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -2, -4, ...].

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u*w)^2 * (u + 4*w)^2 + v^2 * (v*v - 2*u*w)^2 - 2*u*w * (u + 2*v) * (v + 2*w) * (v*v + 4*u*w).

%F If b(n) = A253193(n) - (1 + sqrt(5))/2 * a(n) then b() is multiplicative with b(23^e) = 1, otherwise b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)).

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (23 t)) = 23 (t/i)^2 f(t) where q = exp(2 Pi i t).

%F G.f.: x^2 * Product_{k>0} ((1 - x^k) * (1 - x^(23*k)))^2.

%F Convolution square of A030199.

%e G.f. = q^2 - 2*q^3 - q^4 + 2*q^5 + q^6 + 2*q^7 - 2*q^8 - 2*q^10 - 2*q^11 + ...

%t a[ n_] := SeriesCoefficient[ q^2 (QPochhammer[ q] QPochhammer[ q^23])^2, {q, 0, n}];

%o (PARI) {a(n) = local(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^23 + A))^2, n))};

%o (Sage) CuspForms( Gamma0(23), 2, prec=78).1;

%o (Magma) Basis( CuspForms( Gamma0(23), 2), 78) [2];

%Y Cf. A030199, A253193.

%K sign

%O 2,2

%A _Michael Somos_, Nov 25 2013