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

0, 1, 7, 45, 287, 1821, 11487, 72045, 449407, 2789181, 17230367, 105996045, 649630527, 3968504541, 24174772447, 146908944045, 890924667647, 5393590283901, 32604530573727, 196853323284045, 1187295678104767, 7154833690143261

Offset: 0

Author

Wolfdieter Lang, Oct 28 2011

Keywords

• nonn
• easy

Comments

For the e.g.f.s and o.g.f.s of such alternating power sums see A196847 (even case) and A196848 (odd case).

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Crossrefs

Cf. A000225, A083323, 2*A053154, A198628.

Programs

Maple

A198629 := proc(n)
    (-3^n+4^n-1+2^n-5^n+6^n)/3 ;
end proc:
seq(A198629(n),n=0..20) ; # R. J. Mathar, May 11 2022

Mathematica

Table[Total[Times@@@Partition[Riffle[Range[6]^n,{-1,1},{2,-1,2}],2]]/3,{n,0,30}] (* Harvey P. Dale, Jul 17 2016 *)

Formula

a(n)=sum(((-1)^j)*j^n,j=1..6)/3, n>=0.

E.g.f.: sum(((-1)^j)*exp(j*x),j=1..6)/3 = exp(x)*(exp(6*x)-1)/(3*(exp(x)+1)).

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)/

product(1-j*x,j=1..6). See A196847 for a formula for the coefficients of the numerator polynomial.