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A198630 Alternating sums of powers of 1,2,...,7.

%I A198630 #16 May 11 2022 07:15:23
%S A198630 1,4,28,208,1540,11344,83188,607408,4416580,31986064,230784148,
%T A198630 1659338608,11892395620,84983496784,605698755508,4306834677808,
%U A198630 30560156566660
%N A198630 Alternating sums of powers of 1,2,...,7.
%C A198630 For the e.g.f.s and o.g.f.s of such alternating power sums see A196847 (even case) and A196848 (odd case).
%H A198630 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (28,-322,1960,-6769,13132,-13068,5040).
%F A198630 a(n)=sum(((-1)^(j+1))*j^n,j=1..7), n>=0.
%F A198630 E.g.f.: sum(((-1)^(j+1))*exp(j*x),j=1..7)= exp(x)*
%F A198630   (1+exp(7*x))/(1+exp(x)).
%F A198630 O.g.f: sum(((-1)^(j+1))/(1-j*x),j=1..7) = (1-24*x+238*x^2-1248*x^3+3661*x^4-5736*x^5+3828*x^6)/
%F A198630   product(1-j*x,j=1..7). See A196848 for a formula for the coefficients of the numerator polynomial.
%e A198630 a(2) = 1^2-2^2+3^2-4^2+5^2-6^2+7^2 = 28.
%p A198630 A198630 := proc(n)
%p A198630     3^n-4^n+1-2^n+5^n-6^n+7^n ;
%p A198630 end proc:
%p A198630 seq(A198630(n),n=0..20) ; # _R. J. Mathar_, May 11 2022
%o A198630 (PARI) a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 5040,-13068,13132,-6769,1960,-322,28]^n*[1;4;28;208;1540;11344;83188])[1,1] \\ _Charles R Greathouse IV_, Jul 06 2017
%Y A198630 Cf. A000225, A083323, 2*A053154, A198628, 3*A198629.
%K A198630 nonn,easy
%O A198630 0,2
%A A198630 _Wolfdieter Lang_, Oct 28 2011