%I #52 Sep 08 2022 08:46:11
%S 1,126,1932,12138,47544,140070,341796,730002,1412208,2531214,4270140,
%T 6857466,10572072,15748278,22780884,32130210,44327136,59978142,
%U 79770348,104476554,134960280,172180806,217198212,271178418
%N Second differences of seventh powers (A001015).
%H Luciano Ancora, <a href="/A255177/b255177.txt">Table of n, a(n) for n = 0..1000</a>
%H Luciano Ancora, <a href="https://upload.wikimedia.org/wikipedia/commons/f/fe/Sum_of_powers.pdf">Sums of powers of positive integers and their recurrence relations</a>, section 0.5.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F G.f.: (1 + 120*x + 1191*x^2 + 2416*x^3 + 1191*x^4 + 120*x^5 + x^6)/(1 - x)^6.
%F a(n) = 14*n*(3*n^4 + 5*n^2 + 1) for n>0, a(0)=1.
%F a(n) = A022523(n)-A022523(n-1). - _R. J. Mathar_, Jul 16 2015
%e Second differences: 1, 126, 1932, 12138, 47544, ... (this sequence)
%e First differences: 1, 127, 2060, 14324, 63801, ... (A152726)
%e ----------------------------------------------------------------------
%e The seventh powers: 1, 128, 2187, 16384, 78125, ... (A001015)
%e ----------------------------------------------------------------------
%e First partial sums: 1, 129, 2316, 18700, 96825, ... (A000541)
%e Second partial sums: 1, 130, 2446, 21146, 117971, ... (A250212)
%e Third partial sums: 1, 131, 2577, 23723, 141694, ... (A254641)
%e Fourth partial sums: 1, 132, 2709, 26432, 168126, ... (A254646)
%e Fifth partial sums: 1, 133, 2842, 29274, 197400, ... (A254684)
%t Join[{1}, Table[14 n (3 n^4 + 5 n^2 + 1), {n, 1, 30}], {n, 0, 24}] (* or *)
%t CoefficientList[Series[(1 + 120 x + 1191 x^2 + 2416 x^3 + 1191 x^4 + 120 x^5 + x^6)/(1 - x)^6, {x, 0, 22}], x]
%o (Magma) [1] cat [14*(-1+n)*(9-22*n+23*n^2-12*n^3+3*n^4): n in [2..30]]; // _Vincenzo Librandi_, Mar 12 2015
%Y Cf. A000541, A001015, A152726, A250212, A254641, A254646, A254684, A255178, A255179.
%K nonn,easy
%O 0,2
%A _Luciano Ancora_, Feb 21 2015
%E Edited by _Bruno Berselli_, Mar 19 2015