A198630 Alternating sums of powers of 1,2,...,7.
1, 4, 28, 208, 1540, 11344, 83188, 607408, 4416580, 31986064, 230784148, 1659338608, 11892395620, 84983496784, 605698755508, 4306834677808, 30560156566660
Offset: 0
Examples
a(2) = 1^2-2^2+3^2-4^2+5^2-6^2+7^2 = 28.
Links
- Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).
Programs
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Maple
A198630 := proc(n) 3^n-4^n+1-2^n+5^n-6^n+7^n ; end proc: seq(A198630(n),n=0..20) ; # R. J. Mathar, May 11 2022
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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
Formula
a(n)=sum(((-1)^(j+1))*j^n,j=1..7), n>=0.
E.g.f.: sum(((-1)^(j+1))*exp(j*x),j=1..7)= exp(x)*
(1+exp(7*x))/(1+exp(x)).
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)/
product(1-j*x,j=1..7). See A196848 for a formula for the coefficients of the numerator polynomial.
Comments