A254145 a(n) = 1*7^n + 2*6^n + 3*5^n + 4*4^n + 5*3^n + 6*2^n + 7*1^n.

28, 84, 336, 1596, 8400, 47244, 278256, 1695036, 10592400, 67518444, 437200176, 2867080476, 18997064400, 126948964044, 854359702896, 5783851121916, 39350309552400, 268842017200044, 1843254419626416, 12675940450459356

Offset

0,1

Comments

This is the sequence of seventh terms of "second partial sums of m-th powers".

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Luciano Ancora, Demonstration of formula

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

Formula

G.f.: -4*(17316*x^6 - 32926*x^5 + 24199*x^4 - 8911*x^3 + 1750*x^2 - 175*x + 7) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)). - Colin Barker, Jan 26 2015

From Peter Bala, Jan 31 2016: (Start)

a(n) = (x + 1)*( Bernoulli(n + 1, x + 1) - Bernoulli(n + 1, 1) )/(n + 1) - ( Bernoulli(n + 2, x + 1) - Bernoulli(n + 2, 1) )/(n + 2) at x = 7.

a(n) = (1/6!)*Sum_{k = 0..n} (-1)^(k+n)*(k + 8)!*Stirling2(n,k)/ ((k + 1)*(k + 2)). (End)

Maple

seq(add(i*(8 - i)^n, i = 1..7), n = 0..20); # Peter Bala, Jan 31 2017

Mathematica

Table[6 2^n + 4 4^n + 5 3^n + 2 6^n + 3 5^n + 7^n + 7, {n, 0, 25}] (*  *)
LinearRecurrence[{28, -322, 1960, -6769, 13132, -13068, 5040}, {28, 84, 336, 1596, 8400, 47244, 278256}, 30] (* or *) Table[Total[ Range[ 7]Range[ 7, 1, -1]^n], {n, 0, 20}] (* Harvey P. Dale, Jun 21 2016 *)

Prog

(PARI) Vec(-4*(17316*x^6 -32926*x^5 +24199*x^4 -8911*x^3 +1750*x^2 -175*x +7) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1))  + O(x^100)) \\ Colin Barker, Jan 26 2015

Crossrefs

Cf. A052548, A254028, A254030, A254031, A254144, A254146.

Sequence in context: A165009 A179790 A306427 * A271735 A280885 A042542

Adjacent sequences: A254142 A254143 A254144 * A254146 A254147 A254148

Keyword

nonn,easy

Author

Luciano Ancora, Jan 26 2015

Status

approved