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A198628
Alternating sums of powers for 1,2,3,4 and 5.
4
1, 3, 15, 81, 435, 2313, 12195, 63801, 331395, 1710153, 8775075, 44808921, 227890755, 1155180393, 5839846755, 29458152441, 148335904515, 745888593033, 3746364947235, 18799770158361, 94271405748675, 472449569948073, 2366624981836515, 11850654345690681, 59323452211439235
OFFSET
0,2
COMMENTS
See A196848 for the e.g.f.s and o.g.f.s of such sequences for the numbers 1,2,...,2*n+1, and A196847
for the numbers 1,2,...,2*n.
FORMULA
a(n) = sum(((-1)^(j+1))*j^n,j=1..5) = 1-2^n+3^n-4^n+5^n.
E.g.f.: sum(((-1)^(j+1))*exp(j*x),j=1..5) =
exp(x)*(1+exp(5*x))/(1+exp(x)).
O.g.f.: sum(((-1)^(j+1))/(1-j*x),j=1..5) =
(1-12*x+55*x^2-114*x^3+94*x^4)/product(1-j*x,j=1..5).
A formula for the numbers of the numerator polynomial is given in A196848.
MAPLE
A198628 := proc(n)
3^n-4^n+1-2^n+5^n ;
end proc:
seq(A198628(n), n=0..20) ; # R. J. Mathar, May 11 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Oct 27 2011
STATUS
approved