A254145 - OEIS

%I #36 Sep 27 2024 13:40:19

%S 28,84,336,1596,8400,47244,278256,1695036,10592400,67518444,437200176,

%T 2867080476,18997064400,126948964044,854359702896,5783851121916,

%U 39350309552400,268842017200044,1843254419626416,12675940450459356

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

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

%H Colin Barker, <a href="/A254145/b254145.txt">Table of n, a(n) for n = 0..1000</a>

%H Luciano Ancora, <a href="/A254145/a254145_2.pdf">Demonstration of formula</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (28,-322,1960,-6769,13132,-13068,5040).

%F 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

%F From _Peter Bala_, Jan 31 2016: (Start)

%F 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.

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

%p seq(add(i*(8 - i)^n, i = 1..7), n = 0..20); # _Peter Bala_, Jan 31 2017

%t Table[6 2^n + 4 4^n + 5 3^n + 2 6^n + 3 5^n + 7^n + 7, {n, 0, 25}] (*  *)

%t 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 *)

%o (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

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

%K nonn,easy

%O 0,1

%A _Luciano Ancora_, Jan 26 2015