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%I #11 May 11 2022 07:13:33
%S 0,1,7,45,287,1821,11487,72045,449407,2789181,17230367,105996045,
%T 649630527,3968504541,24174772447,146908944045,890924667647,
%U 5393590283901,32604530573727,196853323284045,1187295678104767,7154833690143261
%N Alternating sums of powers of 1,2,...,6, divided by 3.
%C For the e.g.f.s and o.g.f.s of such alternating power sums see A196847 (even case) and A196848 (odd case).
%H Harvey P. Dale, <a href="/A198629/b198629.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (21,-175,735,-1624,1764,-720).
%F a(n)=sum(((-1)^j)*j^n,j=1..6)/3, n>=0.
%F E.g.f.: sum(((-1)^j)*exp(j*x),j=1..6)/3 = exp(x)*(exp(6*x)-1)/(3*(exp(x)+1)).
%F O.g.f.: sum(((-1)^j)/(1-j*x),j=1..6)/3 = x*(1-14*x+73*x^2-168*x^3+148*x^4)/
%F product(1-j*x,j=1..6). See A196847 for a formula for the coefficients of the numerator polynomial.
%p A198629 := proc(n)
%p (-3^n+4^n-1+2^n-5^n+6^n)/3 ;
%p end proc:
%p seq(A198629(n),n=0..20) ; # _R. J. Mathar_, May 11 2022
%t Table[Total[Times@@@Partition[Riffle[Range[6]^n,{-1,1},{2,-1,2}],2]]/3,{n,0,30}] (* _Harvey P. Dale_, Jul 17 2016 *)
%Y Cf. A000225, A083323, 2*A053154, A198628.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Oct 28 2011
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