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Convolution inverse of A007267.
1

%I #31 May 07 2023 07:06:48

%S 1,-104,6444,-311744,13018830,-493025760,17411253944,-583472867840,

%T 18770817643749,-584450497233840,17716721171780388,

%U -525192444572011776,15276991910654781638,-437229195695756884672,12338641730218147891560,-343932138212987023388672

%N Convolution inverse of A007267.

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

%H Vaclav Kotesovec, <a href="/A286329/b286329.txt">Table of n, a(n) for n = 1..720</a>

%F a(n) ~ -(-1)^n * Gamma(3/4)^16 * exp(Pi*n) * n^3 / (24*Pi^4). - _Vaclav Kotesovec_, Jun 03 2018

%F 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

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

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

%t 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 *)

%t 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 *)

%o (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

%Y Cf. A007267.

%K sign

%O 1,2

%A _Seiichi Manyama_, May 07 2017