A232506 Expansion of (eta(q) * eta(q^23))^2 in powers of q.

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, 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, 4, 4, -8, -3, 4, -2, 6, 2, 2, -2, -2, 4, 2, -4, -4, 2, -2, 2, -8

Offset

2,2

Links

G. C. Greubel, Table of n, a(n) for n = 2..1000

John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472, 2015. See Eq. (B.31).

Formula

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

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, ...].

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).

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)).

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).

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

Convolution square of A030199.

Example

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 + ...

Mathematica

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

Prog

(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))};
(Sage) CuspForms( Gamma0(23), 2, prec=78).1;
(Magma) Basis( CuspForms( Gamma0(23), 2), 78) [2];

Crossrefs

Cf. A030199, A253193.

Sequence in context: A369239 A002107 A208845 * A133099 A006571 A243906

Adjacent sequences: A232503 A232504 A232505 * A232507 A232508 A232509

Keyword

sign

Author

Michael Somos, Nov 25 2013

Status

approved