login
A035174
Ramanujan's tau function (or tau numbers (A000594)) for 2^n.
3
1, -24, -1472, 84480, 987136, -196706304, 2699296768, 338071388160, -13641873096704, -364965248630784, 36697722069188608, -133296500464680960, -71957818786545926144, 1999978883828768833536, 99370119662955604738048
OFFSET
0,2
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.
MATHEMATICA
Table[ RamanujanTau[2^n], {n, 0, 14}]
PROG
(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
(PARI) Vec(1/(1+24*x+2048*x^2)+O(x^99)) \\ Charles R Greathouse IV, Apr 28 2013
(Perl) use ntheory ":all"; say "$_ ", ramanujan_tau(1 << $_) for 0..63; # Dana Jacobsen, Sep 05 2015
CROSSREFS
Sequence in context: A010797 A260784 A099060 * A288955 A203973 A063885
KEYWORD
sign,easy
AUTHOR
Robert G. Wilson v, Jan 04 2003
STATUS
approved