A198628 Alternating sums of powers for 1,2,3,4 and 5.

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.

Links

Table of n, a(n) for n=0..24.

Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

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

Cf. A083323, A196847, A196848, A196837.

Sequence in context: A253774 A003448 A229841 * A233020 A246020 A084120

Adjacent sequences: A198625 A198626 A198627 * A198629 A198630 A198631

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 27 2011

Status

approved