A035174 Ramanujan's tau function (or tau numbers (A000594)) for 2^n.
1, -24, -1472, 84480, 987136, -196706304, 2699296768, 338071388160, -13641873096704, -364965248630784, 36697722069188608, -133296500464680960, -71957818786545926144, 1999978883828768833536, 99370119662955604738048
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..604
- Index entries for linear recurrences with constant coefficients, signature (-24,-2048).
Programs
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Mathematica
Table[ RamanujanTau[2^n], {n, 0, 14}] -
PARI
a(n)=sum(j=0,n\2,(-1)^j*binomial(n-j, n-2*j)*2^(11*j)*(-24)^(n-2*j)) \\ Charles R Greathouse IV, Apr 28 2013
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PARI
Vec(1/(1+24*x+2048*x^2)+O(x^99)) \\ Charles R Greathouse IV, Apr 28 2013
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Perl
use ntheory ":all"; say "$ ",ramanujan_tau(1 << $) for 0..63; # Dana Jacobsen, Sep 05 2015
Formula
G.f.: 1/(1 + 24x + 2048x^2). Proof by Robin Chapman: Follows from the formula tau(p^{n+2}) = tau(p)tau(p^{n+1}) - p^11 tau(p^n) for prime p, which comes from the theory of Hecke operators on modular forms. The p = 2 case gives a recurrence for tau(2^n) leading immediately to the g.f.