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

Author

Wolfdieter Lang, Oct 27 2011

Keywords

• nonn
• easy

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

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

Crossrefs

Cf. A083323, A196847, A196848, A196837.

Programs

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

Mathematica

LinearRecurrence[{15,-85,225,-274,120},{1,3,15,81,435},30] (* Harvey P. Dale, Dec 30 2024 *)

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.