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A308492 Expansion of Sum_{i>=1} mu(i) * x^i * Product_{j>=1} (1 - x^(i*j))^24.

%I A308492 #12 Feb 16 2025 08:33:55
%S A308492 1,-25,251,-1448,4829,-6275,-16745,85952,-113895,-120725,534611,
%T A308492 -363448,-577739,418625,1212079,902656,-6905935,2847375,10661419,
%U A308492 -6992392,-4202995,-13365275,18643271,21573952,-25504055,14443475,-73165437,24246760,128406629,-30301975
%N A308492 Expansion of Sum_{i>=1} mu(i) * x^i * Product_{j>=1} (1 - x^(i*j))^24.
%C A308492 Moebius transform of A000594.
%H A308492 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TauFunction.html">Tau Function</a>
%F A308492 a(n) = Sum_{d|n} mu(n/d)*A000594(d).
%e A308492 G.f. = x - 25*x^2 + 251*x^3 - 1448*x^4 + 4829*x^5 - 6275*x^6 - 16745*x^7 + 85952*x^8 - 113895*x^9 - 120725*x^10 + ...
%t A308492 nmax = 30; CoefficientList[Series[Sum[MoebiusMu[i] x^i Product[(1 - x^(i j))^24, {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A308492 a[n_] := Sum[MoebiusMu[n/d] RamanujanTau[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 30}]
%o A308492 (PARI) a(n) = sumdiv(n, d, moebius(n/d)*ramanujantau(d)); \\ _Michel Marcus_, Jun 01 2019
%Y A308492 Cf. A000594, A008683, A034777, A034778.
%K A308492 sign,mult
%O A308492 1,2
%A A308492 _Ilya Gutkovskiy_, May 31 2019