|
%I #16 Sep 08 2022 08:45:23
%S 1,36,1793,24604,167481,756836,2620201,7526268,18831569,42374116,
%T 87654321,169343516,309160393,538155684,899445401,1451432956,
%U 2271560481,3460629668,5147732449,7495831836,10708033241,15034586596,20780659593
%N a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7.
%C 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + 7*n^6 + 8*n^7 is the derivative of 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 = (x^9 - 1)/(x-1).
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8, -28, 56, -70, 56, -28, 8, -1).
%F G.f.: (1+28*x+1533*x^2+11212*x^3+18907*x^4+7956*x^5+679*x^6+4*x^7)/(x-1)^8. - _R. J. Mathar_, Dec 21 2010
%F a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), with a(0)=1, a(1)=36, a(2)=1793, a(3)=24604, a(4)=167481, a(5)=756836, a(6)=2620201, a(7)=7526268. - _Harvey P. Dale_, Jul 16 2014
%e 1 + 2*8 + 3*8^2 + 4*8^3 + 5*8^4 + 6*8^5 + 7*8^6 + 8*8^7 = 18831569 = 173 * 199 * 547.
%e 1 + 2*26 + 3*26^2 + 4*26^3 + 5*26^4 + 6*26^5 + 7*26^6 + 8*26^7 = 66490537361 is prime, the smallest prime in the sequence.
%t Join[{1},Table[Total[Table[p*n^(p-1),{p,8}]],{n,30}]] (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,36,1793,24604,167481,756836,2620201,7526268},30] (* _Harvey P. Dale_, Jul 16 2014 *)
%o (Magma) [1+2*n+3*n^2+4*n^3+5*n^4+6*n^5+7*n^6+8*n^7: n in [1..43]] // _Vincenzo Librandi_, Dec 21 2010
%Y Cf. A000012, A005408, A056109, A056578, A056579.
%K easy,nonn
%O 0,2
%A _Jonathan Vos Post_, Jan 14 2006
|
