%I #33 Jan 26 2022 02:29:38
%S 1,22,198,1134,4884,17226,52338,141570,348777,795652,1701700,3444948,
%T 6651216,12321804,22011804,38073948,63985977,104782986,167620090,
%U 262495090,403165620,608300550,902911230,1320114510,1903286385,2708672616,3808530792,5294887048
%N Sixth partial sums of fourth powers (A000583).
%H Luciano Ancora, <a href="/A254470/b254470.txt">Table of n, a(n) for n = 1..1000</a>
%H Luciano Ancora, <a href="/A254640/a254640_1.pdf">Partial sums of m-th powers with Faulhaber polynomials</a>.
%H Luciano Ancora, <a href="/A254647/a254647_2.pdf">Pascal’s triangle and recurrence relations for partial sums of m-th powers</a>.
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
%F G.f.: (-x - 11*x^2 - 11*x^3 - x^4)/(- 1 + x)^11.
%F a(n) = n*(1 + n)*(2 + n)*(3 + n)^2*(4 + n)*(5 + n)*(6 + n)*(1 + 12*n + 2*n^2)/302400.
%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) + n^4.
%F Sum_{n>=1} 1/a(n) = 3320303/2601 + 1400*Pi^2/17 + (8960/17)*sqrt(2/17)*Pi*cot(sqrt(17/2)*Pi). - _Amiram Eldar_, Jan 26 2022
%e First differences: 1, 15, 65, 175, 369, 671, ... (A005917)
%e -------------------------------------------------------------------------
%e The fourth powers: 1, 16, 81, 256, 625, 1296, ... (A000583)
%e -------------------------------------------------------------------------
%e First partial sums: 1, 17, 98, 354, 979, 2275, ... (A000538)
%e Second partial sums: 1, 18, 116, 470, 1449, 3724, ... (A101089)
%e Third partial sums: 1, 19, 135, 605, 2054, 5778, ... (A101090)
%e Fourth partial sums: 1, 20, 155, 760, 2814, 8592, ... (A101091)
%e Fifth partial sums: 1, 21, 176, 936, 3750, 12342, ... (A254681)
%e Sixth partial sums: 1, 22, 198,1134, 4884, 17226, ... (this sequence)
%t Table[n (1 + n) (2 + n) (3 + n)^2 (4 + n) (5 + n) (6 + n) (1 + 12 n + 2 n^2)/302400,{n, 25}] (* or *) CoefficientList[Series[(- 1 - 11 x - 11 x^2 - x^3)/(- 1 + x)^11, {x, 0, 24}], x]
%t Nest[Accumulate,Range[30]^4,6] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,22,198,1134,4884,17226,52338,141570,348777,795652,1701700},30] (* _Harvey P. Dale_, Apr 23 2016 *)
%o (Magma) [n*(1+n)*(2+n)*(3+n)^2*(4+n)*(5+n)*(6+n)*(1+12*n+ 2*n^2)/302400: n in [1..30]]; // _Vincenzo Librandi_, Feb 15 2015
%o (PARI) vector(50,n,n*(1 + n)*(2 + n)*(3 + n)^2*(4 + n)*(5 + n)*(6 + n)*(1 + 12*n + 2*n^2)/302400) \\ _Derek Orr_, Feb 19 2015
%Y Cf. A000538, A000583, A005917, A101089, A101090, A101091, A254469, A254471, A254472, A254681.
%K nonn,easy
%O 1,2
%A _Luciano Ancora_, Feb 15 2015