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 A254146 a(n) = 1*8^n + 2*7^n + 3*6^n + 4*5^n + 5*4^n + 6*3^n + 7*2^n + 8*1^n. 7

%I

%S 36,120,540,2892,17172,109020,725220,4992492,35277012,254402940,

%T 1864757700,13850340492,103996064052,787943896860,6015370201380,

%U 46217575406892,357036252710292,2770979252910780,21591510288112260,168818732978719692,1323861500735007732

%N a(n) = 1*8^n + 2*7^n + 3*6^n + 4*5^n + 5*4^n + 6*3^n + 7*2^n + 8*1^n.

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

%H Colin Barker, <a href="/A254146/b254146.txt">Table of n, a(n) for n = 0..1000</a>

%H Luciano Ancora, <a href="/A254146/a254146_1.pdf">Demonstration of formulas</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (36,-546,4536,-22449,67284,-118124,109584,-40320).

%F G.f.: -12*(55308*x^7 - 113262*x^6 + 92327*x^5 - 39312*x^4 + 9527*x^3 - 1323*x^2 + 98*x -3) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(7*x-1)*(8*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 = 8.

%F a(n) = 1/7!*Sum_{k = 0..n} (-1)^(k+n)*(k + 9)!*Stirling2(n,k)/ ((k + 1)*(k + 2)). (End)

%F a(n) = 36*a(n-1)-546*a(n-2)+4536*a(n-3)-22449*a(n-4)+67284*a(n-5)-118124*a(n-6)+109584*a(n-7)-40320*a(n-8). - _Wesley Ivan Hurt_, May 24 2021

%p seq(add(i*(9 - i)^n, i = 1..8), n = 0..20); # _Peter Bala_, Jan 31 2017

%t Table[7 2^n + 5 4^n + 8^n + 6 3^n + 3 6^n + 4 5^n + 2 7^n + 8, {n, 0, 30}] (* _Vincenzo Librandi_, Jan 28 2015 *)

%o (PARI) vector(30, n, n--; 7*2^n + 5*4^n + 8^n + 6*3^n + 3*6^n + 4*5^n + 2*7^n + 8) \\ _Colin Barker_, Jan 28 2015

%o (MAGMA) [7*2^n+5*4^n+8^n+6*3^n+3*6^n+4*5^n+2*7^n+8: n in [0..30]]; // _Vincenzo Librandi_, Jan 28 2015

%Y Cf. A052548, A254028, A254030, A254031, A254144, A254145.

%K nonn,easy

%O 0,1

%A _Luciano Ancora_, Jan 27 2015

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Last modified July 26 14:45 EDT 2021. Contains 346294 sequences. (Running on oeis4.)