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A198630
Alternating sums of powers of 1,2,...,7.
1
1, 4, 28, 208, 1540, 11344, 83188, 607408, 4416580, 31986064, 230784148, 1659338608, 11892395620, 84983496784, 605698755508, 4306834677808, 30560156566660
OFFSET
0,2
COMMENTS
For the e.g.f.s and o.g.f.s of such alternating power sums see A196847 (even case) and A196848 (odd case).
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.
EXAMPLE
a(2) = 1^2-2^2+3^2-4^2+5^2-6^2+7^2 = 28.
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
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Oct 28 2011
STATUS
approved