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%I #36 Jan 27 2022 03:09:07
%S 1,37,418,2754,13080,49632,159654,452166,1157013,2724865,5988268,
%T 12410476,24456744,46132152,83740980,146935284,250134753,414416277,
%U 669990046,1059399550,1641605680,2497140360,3734542890,5498322570
%N Fifth partial sums of fifth powers (A000584).
%H Luciano Ancora, <a href="/A254682/b254682.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 - 26*x^2 - 66*x^3 - 26*x^4 - x^5)/(- 1 + x)^11.
%F a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(- 2 + 5*n + n^2)*(9 + 10*n + 2*n^2)/60480.
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) + n^5.
%F Sum_{n>=1} 1/a(n) = 475867/180 - (2560/13)*sqrt(7)*Pi*tan(sqrt(7)*Pi/2) + (210/13)*sqrt(3/11)*Pi*tan(sqrt(33)*Pi/2). - _Amiram Eldar_, Jan 27 2022
%e Fifth differences: 1, 27, 93, 119, 120, (repeat 120) (A101100)
%e Fourth differences: 1, 28, 121, 240, 360, 480, ... (A101095)
%e Third differences: 1, 29, 150, 390, 750, 1230, ... (A101096)
%e Second differences: 1, 30, 180, 570, 1320, 2550, ... (A101098)
%e First differences: 1, 31, 211, 781, 2101, 4651, ... (A022521)
%e -------------------------------------------------------------------------
%e The fifth powers: 1, 32, 243, 1024, 3125, 7776, ... (A000584)
%e -------------------------------------------------------------------------
%e First partial sums: 1, 33, 276, 1300, 4425, 12201, ... (A000539)
%e Second partial sums: 1, 34, 310, 1610, 6035, 18236, ... (A101092)
%e Third partial sums: 1, 35, 345, 1955, 7990, 26226, ... (A101099)
%e Fourth partial sums: 1, 36, 381, 2336, 10326, 36552, ... (A254644)
%e Fifth partial sums: 1, 37, 418, 2754, 13080, 49632, ... (this sequence)
%t Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (- 2 + 5 n + n^2) (9 + 10 n + 2 n^2)/60480, {n,24}] (* or *)
%t CoefficientList[Series[(- 1 - 26 x - 66 x^2 - 26 x^3 - x^4)/(- 1 + x)^11, {x,0,23}], x]
%t Nest[Accumulate,Range[30]^5,5] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,37,418,2754,13080,49632,159654,452166,1157013,2724865,5988268},30] (* _Harvey P. Dale_, Jan 30 2019 *)
%o (PARI) a(n)=n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(-2+5*n+n^2)*(9+10*n+2*n^2)/60480 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A000539, A000584, A022521, A101092, A101095, A101096, A101098, A101099, A101100, A254644, A254681, A254683, A254684.
%K nonn,easy
%O 1,2
%A _Luciano Ancora_, Feb 12 2015
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