%I #24 Sep 08 2022 08:46:11

%S 1,509,18150,204630,1225230,4985070,15717750,41436870,95750430,

%T 200038110,385991430,698516790,1199001390,1968942030,3113936790,

%U 4768039590,7098477630,10310731710,14653979430,20426901270

%N Third differences of ninth powers (A001017).

%H Luciano Ancora, <a href="/A255183/b255183.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F G.f.: (1 + 502*x + 14608*x^2 + 88234*x^3 + 156190*x^4 + 88234*x^5 + 14608*x^6 + 502*x^7 + x^8)/(1 - x)^7.

%F a(n) = 6*(84*n^6 - 252*n^5 + 630*n^4 - 840*n^3 + 756*n^2 - 378*n + 85) for n>1, a(0)=1, a(1)=509.

%e Third differences: 1, 509, 18150, 204630, 1225230, ... (this sequence)

%e Second differences: 1, 510, 18660, 223290, 1448520, ... (A255179)

%e First differences: 1, 511, 19171, 242461, 1690981, ... (A022525)

%e ---------------------------------------------------------------------

%e The ninth powers: 1, 512, 19683, 262144, 1953125, ... (A001017)

%e ---------------------------------------------------------------------

%t Join[{1, 509}, Table[6 (84 n^6 - 252 n^5 + 630 n^4 - 840 n^3 + 756 n^2 - 378 n + 85), {n, 2, 30}]]

%t Join[{1,509},Differences[Range[0,20]^9,3]] (* _Harvey P. Dale_, Apr 24 2015 *)

%o (Magma) [1,509] cat [6*(84*n^6-252*n^5+630*n^4-840*n^3+756*n^2-378*n+85): n in [2..30]]; // _Vincenzo Librandi_, Mar 18 2015

%Y Cf. A001017, A007487, A022525, A253637, A255179, A255182.

%K nonn,easy

%O 0,2

%A _Luciano Ancora_, Mar 18 2015

%E Edited by _Bruno Berselli_, Mar 20 2015