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, 216449743135504, 1530600006189268, 10808335215521008, 76230379263297700, 537081363012854224, 3780565969722557428

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).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Formula

a(n) = Sum_{j=1..7} (-1)^(j+1)*j^n.

E.g.f.: Sum_{j=1..7} ((-1)^(j+1))*exp(j*x) = exp(x)*(1+exp(7*x))/(1+exp(x)).

O.g.f.: Sum_{j=1..7} ((-1)^(j+1))/(1 - j*x).

O.g.f.: (1-24*x+238*x^2-1248*x^3+3661*x^4-5736*x^5+3828*x^6)/(Product_{j=1..7} (1-j*x)). 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
(PARI) a(n) = sum(j=1, 7, (-1)^(j+1)*j^n) \\ Andrew Howroyd, Nov 09 2025

Crossrefs

Cf. A000225, A083323, 2*A053154, A198628, 3*A198629.

Sequence in context: A355354 A019482 A397359 * A246021 A090965 A106258

Adjacent sequences: A198627 A198628 A198629 * A198631 A198632 A198633

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 28 2011

Extensions

a(17) onward from Andrew Howroyd, Nov 09 2025

Status

approved