A051951 - OEIS

%I #15 Jan 29 2017 08:57:26

%S 0,-1,1,-2,3,-4,4,-3,2,-3,6,-8,6,-2,1,-4,7,-8,8,-6,2,-2,8,-11,6,-1,2,

%T -6,10,-12,10,-6,2,0,5,-12,9,-2,4,-10,12,-12,10,-4,-2,0,10,-15,10,-5,

%U 4,-6,10,-10,8,-8,4,-2,12,-20,12,0,-1,-4,7,-10,10,-6,6,-10,16,-20,12,0,-2,-2,6,-10,14,-13,4

%N Second differences of tau(n).

%C a(0) = 0 assuming tau(0) = 0. - _Michael Somos_, Mar 05 2014

%H Robert Israel, <a href="/A051951/b051951.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: -1/x + (1 - x)^2*Sum_{k>=1} x^(k-2)/(1 - x^k). - _Ilya Gutkovskiy_, Jan 29 2017

%e G.f. = - x + x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 4*x^6 - 3*x^7 + 2*x^8 - 3*x^9 + ...

%p Taus:= map(numtheory:-tau, [$0..100]):

%p Taus[3..-1] - 2*Taus[2..-2] + Taus[1..-3]; # _Robert Israel_, Dec 07 2015

%t Join[{0},Differences[DivisorSigma[0,Range[90]],2]] (* _Harvey P. Dale_, Jan 10 2014 *)

%t a[ n_] := SeriesCoefficient[ (QHypergeometricPFQ[ {x, x}, {x^2, x^2}, x, x^2] - 1) / x, {x, 0, n}]; (* _Michael Somos_, Mar 05 2014 *)

%Y Cf. A000005, A051950.

%K sign

%O 0,4

%A _N. J. A. Sloane_, Dec 17 1999