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

0,3

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).

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.

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 *)

Crossrefs

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

Sequence in context: A323140 A236194 A230760 * A190973 A099842 A287811

Adjacent sequences: A198626 A198627 A198628 * A198630 A198631 A198632

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 28 2011

Status

approved