A256552 Expansion of the unique weight 11/2 Gamma1(4) cusp form in powers of q.

1, -2, -8, 16, 20, -36, 0, -32, -75, 220, 104, -128, -44, -392, 0, 256, 232, 474, -536, 320, 168, -1124, 0, -576, 245, 852, 1248, 0, -1668, 2040, 0, -512, -1368, -2632, -560, -1200, 4756, 1428, 0, 3520, 656, -3528, -3224, 1664, -4740, 2168, 0, -2048, 1449

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

Expansion of q * f(-q)^2 * f(-q^2)^7 * f(-q^4)^2 in powers of q where f() is a Ramanujan theta function.

Expansion of eta(q)^2 * eta(q^2)^7 * eta(q^4)^2 in powers of q.

Euler transform of period 4 sequence [ -2, -9, -2, -11, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2^(11/2) (t/i)^(11/2) f(t) where q = exp(2 Pi i t).

G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 - x^(2*k))^7 * (1 - x^(4*k))^2.

a(8*n + 7) = 0. a(4*n) = 16 * a(n).

Example

G.f. = q - 2*q^2 - 8*q^3 + 16*q^4 + 20*q^5 - 36*q^6 - 32*q^8 - 75*q^9 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^2 QPochhammer[ q^2]^7  QPochhammer[ q^4]^2, {q, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^2 + A)^7 * eta(x^4 + A)^2, n))};
(Magma) Basis( CuspForms( Gamma1(4), 11/2), 50)[1];

Crossrefs

Sequence in context: A193219 A213249 A155853 * A031061 A175819 A125259

Adjacent sequences: A256549 A256550 A256551 * A256553 A256554 A256555

Keyword

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Author

Michael Somos, Apr 01 2015

Status

approved