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%I #22 Jun 13 2015 00:55:25
%S 1,71,1205,11075,70295,345857,1409387,4962365,15539750,44192010,
%T 115917118,283828498,654885730,1434717550,3002927770,6035661334,
%U 11699568079,21951176425,39988722875,70920437325,122735050305
%N Seventh partial sums of sixth powers (A001014).
%H Luciano Ancora, <a href="/A254872/b254872.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_14">Index entries for linear recurrences with constant coefficients</a>, signature (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).
%F G.f.: (x + 57*x^2 + 302*x^3 + 302*x^4 + 57*x^5 + x^6)/(- 1 + x)^14.
%F a(n) = (n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 2*n)*(- 49 + 147*n^2 + 42*n^3 + 3*n^4))/51891840.
%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) + n^6.
%e First differences: 1, 63, 665, 3367, 11529, ... (A022522)
%e --------------------------------------------------------------------
%e The sixth powers: 1, 64, 729, 4096, 15625, ... (A001014)
%e --------------------------------------------------------------------
%e First partial sums: 1, 65, 794, 4890, 20515, ... (A000540)
%e Second partial sums: 1, 66, 860, 5750, 26265, ... (A101093)
%e Third partial sums: 1, 67, 927, 6677, 32942, ... (A254640)
%e Fourth partial sums: 1, 68, 995, 7672, 40614, ... (A254645)
%e Fifth partial sums: 1, 69, 1064, 8736, 49350, ... (A254683)
%e Sixth partial sums: 1, 70, 1134, 9870, 59220, ... (A254472)
%e Seventh partial sums: 1, 71, 1205, 11075, 70295, ... (this sequence)
%t Table[(n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) (7 + 2 n) (- 49 + 147 n^2 + 42 n^3 + 3 n^4))/51891840, {n, 21}] (* or *)
%t CoefficientList[Series[(1 + 57 x + 302 x^2 + 302 x^3 + 57 x^4 + x^5)/(- 1 + x)^14, {x, 0, 20}], x]
%Y Cf. A000540, A001014, A022522, A101093, A254472, A254640, A254645, A254683, A254869, A254870, A254871.
%K nonn,easy
%O 1,2
%A _Luciano Ancora_, Feb 17 2015
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