Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #12 May 11 2022 07:10:55
%S 1,3,15,81,435,2313,12195,63801,331395,1710153,8775075,44808921,
%T 227890755,1155180393,5839846755,29458152441,148335904515,
%U 745888593033,3746364947235,18799770158361,94271405748675,472449569948073,2366624981836515,11850654345690681,59323452211439235
%N Alternating sums of powers for 1,2,3,4 and 5.
%C 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
%C for the numbers 1,2,...,2*n.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (15,-85,225,-274,120).
%F a(n) = sum(((-1)^(j+1))*j^n,j=1..5) = 1-2^n+3^n-4^n+5^n.
%F E.g.f.: sum(((-1)^(j+1))*exp(j*x),j=1..5) =
%F exp(x)*(1+exp(5*x))/(1+exp(x)).
%F O.g.f.: sum(((-1)^(j+1))/(1-j*x),j=1..5) =
%F (1-12*x+55*x^2-114*x^3+94*x^4)/product(1-j*x,j=1..5).
%F A formula for the numbers of the numerator polynomial is given in A196848.
%p A198628 := proc(n)
%p 3^n-4^n+1-2^n+5^n ;
%p end proc:
%p seq(A198628(n),n=0..20) ; # _R. J. Mathar_, May 11 2022
%Y Cf. A083323, A196847, A196848, A196837.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Oct 27 2011