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%I #22 Feb 02 2017 03:43:37
%S 45,165,825,4917,32505,229845,1703625,13072917,103008345,828707925,
%T 6779099625,56214660117,471424600185,3990804658005,34053173154825,
%U 292542431786517,2527742384720025,21950298188288085,191434401453597225,1675813243179972117
%N a(n) = 1*9^n + 2*8^n + 3*7^n + 4*6^n + 5*5^n + 6*4^n + 7*3^n + 8*2^n + 9*1^n.
%C This is the sequence of ninth terms of "second partial sums of m-th powers".
%H Colin Barker, <a href="/A254147/b254147.txt">Table of n, a(n) for n = 0..1000</a>
%H Luciano Ancora, <a href="/A254147/a254147.pdf">Demonstration of formulas</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (45,-870,9450,-63273,269325,-723680,1172700,-1026576,362880).
%F G.f.: -3*(2333280*x^8 - 5080464*x^7 + 4500500*x^6 - 2143640*x^5 + 605675*x^4 - 104636*x^3 + 10850*x^2 - 620*x + 15) / ((x - 1)*(2*x - 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(6*x - 1)*(7*x - 1)*(8*x - 1)*(9*x - 1)). - _Colin Barker_, Jan 28 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 = 9.
%F a(n) = (1/8!)*Sum_{k = 0..n} (-1)^(k+n)*(k + 10)!*Stirling2(n,k) /((k + 1)*(k + 2)). (End)
%p seq(add(i*(10-i)^n, i = 1..9), n = 0..20); # _Peter Bala_, Jan 31 2017
%o (PARI) vector(30, n, n--; 8*2^n + 6*4^n + 2*8^n + 7*3^n + 4*6^n + 9^n + 5*5^n + 3*7^n + 9) \\ _Colin Barker_, Jan 28 2015
%Y Cf. A052548, A254028, A254030, A254031, A254144, A254145, A254146.
%K nonn,easy
%O 0,1
%A _Luciano Ancora_, Jan 28 2015
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