|
%I #30 Jan 19 2019 22:19:06
%S 21,56,196,812,3724,18236,93436,494732,2685004,14851676,83384476,
%T 473755052,2717541484,15709845116,91395715516,534498925772,
%U 3139343105164,18504595174556,109397060622556,648335998054892,3850205790608044
%N a(n) = 1*6^n + 2*5^n + 3*4^n + 4*3^n + 5*2^n + 6*1^n.
%C This is the sequence of sixth terms of "second partial sums of m-th powers".
%H Colin Barker, <a href="/A254144/b254144.txt">Table of n, a(n) for n = 0..1000</a>
%H Luciano Ancora, <a href="/A254144/a254144_1.pdf">Demonstration of formulas</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (21,-175,735,-1624,1764,-720).
%F G.f.: -(8028*x^5 - 13916*x^4 + 8939*x^3 - 2695*x^2 + 385*x - 21) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*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 = 6.
%F a(n) = (1/5!)*Sum_{k = 0..n} (-1)^(k+n)*(k + 7)!*Stirling2(n,k)/ ((k + 1)*(k + 2)). (End)
%p seq(add(i*(7 - i)^n, i = 1..6), n = 0..20); # _Peter Bala_, Jan 31 2017
%t Table[5 2^n + 3 4^n + 4 3^n + 2 5^n + 6^n + 6, {n, 0, 25}] (* _Bruno Berselli_, Jan 27 2015 *)
%o (PARI) Vec(-(8028*x^5-13916*x^4+8939*x^3-2695*x^2+385*x-21)/((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)) + O(x^100)) \\ _Colin Barker_, Jan 26 2015
%Y Cf. A052548, A254028, A254030, A254031, A254145, A254146.
%K nonn,easy
%O 0,1
%A _Luciano Ancora_, Jan 26 2015
|
