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A254471 Sixth partial sums of fifth powers (A000584).

%I A254471 #23 Sep 08 2022 08:46:11
%S A254471 1,38,456,3210,16290,65922,225576,677742,1834755,4559620,10547888,
%T A254471 22958364,47415108,93547260,177288240,324223524,574358277,988774554,
%U A254471 1658764600,2718164150,4359769830,6856910190,10591453080,16089775650,24068499975,35492110056
%N A254471 Sixth partial sums of fifth powers (A000584).
%H A254471 Luciano Ancora, <a href="/A254471/b254471.txt">Table of n, a(n) for n = 1..1000</a>
%H A254471 Luciano Ancora, <a href="/A254640/a254640_1.pdf">Partial sums of m-th powers with Faulhaber polynomials</a>
%H A254471 Luciano Ancora, <a href="/A254647/a254647_2.pdf">Pascal’s triangle and recurrence relations for partial sums of m-th powers</a>
%H A254471 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
%F A254471 G.f.: (x + 26*x^2 + 66*x^3 + 26*x^4 + x^5)/(- 1 + x)^12.
%F A254471 a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(-29 + 54*n + 81*n^2 + 24*n^3 + 2*n^4)/665280.
%F A254471 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^5.
%e A254471 First differences:   1, 31, 211,  781,  2101,  4651, ... (A022521)
%e A254471 -------------------------------------------------------------------------
%e A254471 The fifth powers:    1, 32, 243, 1024,  3125,  7776, ... (A000584)
%e A254471 -------------------------------------------------------------------------
%e A254471 First partial sums:  1, 33, 276, 1300,  4425, 12201, ... (A000539)
%e A254471 Second partial sums: 1, 34, 310, 1610,  6035, 18236, ... (A101092)
%e A254471 Third partial sums:  1, 35, 345, 1955,  7990, 26226, ... (A101099)
%e A254471 Fourth partial sums: 1, 36, 381, 2336, 10326, 36552, ... (A254644)
%e A254471 Fifth partial sums:  1, 37, 418, 2754, 13080, 49632, ... (A254682)
%e A254471 Sixth partial sums:  1, 38, 456, 3210, 16290, 65922, ... (this sequence)
%t A254471 Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (- 29 + 54 n + 81 n^2 + 24 n^3 + 2 n^4)/665280, {n, 23}] (* or *) CoefficientList[Series[(1 + 26 x + 66 x^2 + 26 x^3 + x^4)/(- 1 + x)^12, {x, 0, 28}], x]
%o A254471 (Magma) [n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(-29+54*n+ 81*n^2+24*n^3+2*n^4)/665280: n in [1..30]]; // _Vincenzo Librandi_, Feb 15 2015
%o A254471 (PARI) vector(50,n,n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(-29 + 54*n + 81*n^2 + 24*n^3 + 2*n^4)/665280) \\ _Derek Orr_, Feb 19 2015
%Y A254471 Cf. A000539, A000584, A022521, A101092, A101099, A254469, A254470, A254472, A254644, A254682.
%K A254471 nonn,easy
%O A254471 1,2
%A A254471 _Luciano Ancora_, Feb 15 2015