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A064556 Ramanujan's tau function (or tau numbers (A000594)) for 10^n.

%I A064556 #11 Jul 01 2023 16:13:08
%S A064556 1,-115920,37534859200,-30328412970240000,-482606811957501440000,
%T A064556 -2983637890141033828147200000
%N A064556 Ramanujan's tau function (or tau numbers (A000594)) for 10^n.
%H A064556 Seiichi Manyama, <a href="/A064556/b064556.txt">Table of n, a(n) for n = 0..181</a>
%H A064556 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-115920, 124097412800, -11592000000000000, -10000000000000000000000).
%t A064556 Table[ RamanujanTau[10^n], {n, 0, 5}]
%o A064556 (Perl) use bigint; use ntheory ":all"; say "$_ ",ramanujan_tau(10 ** $_) for 0..19; # _Dana Jacobsen_, Sep 05 2015
%o A064556 (PARI) taup(p, e)={
%o A064556     if(e==1,
%o A064556         (65*sigma(p, 11)+691*sigma(p, 5)-691*252*sum(k=1, p-1, sigma(k, 5)*sigma(p-k, 5)))/756
%o A064556     ,
%o A064556         my(t=taup(p, 1));
%o A064556         sum(j=0, e\2,
%o A064556             (-1)^j*binomial(e-j, e-2*j)*p^(11*j)*t^(e-2*j)
%o A064556         )
%o A064556     )
%o A064556 };
%o A064556 a(n)=taup(5,n)*taup(2,n) \\ _Charles R Greathouse IV_, Sep 06 2015
%Y A064556 Cf. A000594, A035174.
%K A064556 sign
%O A064556 0,2
%A A064556 _Robert G. Wilson v_, Jan 04 2003