cp's OEIS Frontend

A198629 Alternating sums of powers of 1,2,...,6, divided by 3.

%I A198629 #11 May 11 2022 07:13:33
%S A198629 0,1,7,45,287,1821,11487,72045,449407,2789181,17230367,105996045,
%T A198629 649630527,3968504541,24174772447,146908944045,890924667647,
%U A198629 5393590283901,32604530573727,196853323284045,1187295678104767,7154833690143261
%N A198629 Alternating sums of powers of 1,2,...,6, divided by 3.
%C A198629 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 A198629 Harvey P. Dale, <a href="/A198629/b198629.txt">Table of n, a(n) for n = 0..1000</a>
%H A198629 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (21,-175,735,-1624,1764,-720).
%F A198629 a(n)=sum(((-1)^j)*j^n,j=1..6)/3, n>=0.
%F A198629 E.g.f.: sum(((-1)^j)*exp(j*x),j=1..6)/3 = exp(x)*(exp(6*x)-1)/(3*(exp(x)+1)).
%F A198629 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 A198629   product(1-j*x,j=1..6). See A196847 for a formula for the coefficients of the numerator polynomial.
%p A198629 A198629 := proc(n)
%p A198629     (-3^n+4^n-1+2^n-5^n+6^n)/3 ;
%p A198629 end proc:
%p A198629 seq(A198629(n),n=0..20) ; # _R. J. Mathar_, May 11 2022
%t A198629 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 A198629 Cf. A000225, A083323, 2*A053154, A198628.
%K A198629 nonn,easy
%O A198629 0,3
%A A198629 _Wolfdieter Lang_, Oct 28 2011