A286329 Convolution inverse of A007267.

1, -104, 6444, -311744, 13018830, -493025760, 17411253944, -583472867840, 18770817643749, -584450497233840, 17716721171780388, -525192444572011776, 15276991910654781638, -437229195695756884672, 12338641730218147891560, -343932138212987023388672

Offset

1,2

References

T. Gannon, Moonshine Beyond the Monster, Cambridge, 2006; see p. 433.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..720

Formula

a(n) ~ -(-1)^n * Gamma(3/4)^16 * exp(Pi*n) * n^3 / (24*Pi^4). - Vaclav Kotesovec, Jun 03 2018

Expansion of A/(q*(1+64*A)^2), with A = (eta(q^2)/eta(q))^24, in powers of q. - G. C. Greubel, Jun 19 2018

Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/648, validated to 900 digits. - Simon Plouffe, May 06 2023

Example

G.f. = q - 104*q^2 + 6444*q^3 - 311744*q^4 + 13018830*q^5 - 493025760*q^6 + ...

Mathematica

nmax = 20; CoefficientList[Series[1/(128*x + Product[1/(1 + x^k)^24, {k, 1, nmax}] + 4096*x^2*Product[(1 + x^k)^24, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 03 2018 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; x2:= (eta[q^2]/eta[q])^24; a:= CoefficientList[Series[x2/(1 + 64*x2)^2/q, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 19 2018 *)

Prog

(PARI) q='q+O('q^50); A = q*(eta(q^2)/eta(q))^24; Vec(A/(q*(1+64*A)^2)) \\ G. C. Greubel, Jun 19 2018

Crossrefs

Cf. A007267.

Sequence in context: A250668 A035811 A004393 * A164759 A206013 A187700

Adjacent sequences: A286326 A286327 A286328 * A286330 A286331 A286332

Keyword

sign

Author

Seiichi Manyama, May 07 2017

Status

approved