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A254643 Third partial sums of ninth powers (A001017). 3

%I #49 Sep 08 2022 08:46:11

%S 1,515,21225,324275,2862790,17714466,85232910,339635850,1168343775,

%T 3571356685,9906622271,25333920885,60457751900,135939162100,

%U 290221510860,592024274916,1159935330765,2192313968775,4011847886725,7130537084615

%N Third partial sums of ninth powers (A001017).

%H Luciano Ancora, <a href="/A254643/b254643.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="/A254643/a254643_2.pdf"> Pascal’s triangle and recurrence relations for partial sums of m-th powers </a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

%F G.f.: x*(1 +502*x +14608*x^2 +88234*x^3 +156190*x^4 +88234*x^5 +14608*x^6 +502*x^7 +x^8)/(1-x)^13.

%F a(n) = n*(1+n)*(2+n)*(3+n)*(-50 +84*n +127*n^2 -204*n^3 -97*n^4 +126*n^5 +98*n^6 +24*n^7 +2*n^8)/2640.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + n^9.

%e First differences: 1, 511, 19171, 242461, 1690981, ... (A022525)

%e ------------------------------------------------------------------------

%e The ninth powers: 1, 512, 19683, 262144, 1953125, ... (A001017)

%e ------------------------------------------------------------------------

%e First partial sums: 1, 513, 20196, 282340, 2235465, ... (A007487)

%e Second partial sums: 1, 514, 20710, 303050, 2538515, ... (A253637)

%e Third partial sums: 1, 515, 21225, 324275, 2862790, ... (this sequence)

%p seq(binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110, n=1..30); # _G. C. Greubel_, Aug 28 2019

%t Table[n(1+n)(2+n)(3+n)(-50 +84n +127n^2 -204n^3 -97n^4 +126n^5 +98n^6 +24n^7 +2n^8)/2640, {n, 20}] (* or *)

%t CoefficientList[Series[(1 +502x +14608x^2 +88234x^3 +156190x^4 +88234x^5 +14608x^6 +502x^7 +x^8)/(1-x)^13, {x, 0, 19}], x] (* Ancora *)

%t Accumulate[Accumulate[Accumulate[Range[10]^9]]] (* _Alonso del Arte_, Feb 09 2015 *)

%o (PARI) vector(30, n, m=n+3; binomial(m,4)*(2*(n*m)^4 -10*(n*m)^3 +11*(n*m)^2 +28*(n*m) -50)/110) \\ _G. C. Greubel_, Aug 28 2019

%o (Magma) [Binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110: n in [1..30]]; // _G. C. Greubel_, Aug 28 2019

%o (Sage) [binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110 for n in (1..30)] # _G. C. Greubel_, Aug 28 2019

%o (GAP) List([1..30], n-> Binomial(n+3,4)*(2*n^8 +24*n^7 +98*n^6 +126*n^5 -97*n^4 -203*n^3 +127*n^2 +84*n -50)/110); # _G. C. Greubel_, Aug 28 2019

%Y Cf. A001017, A007487, A022525, A253637.

%K nonn,easy

%O 1,2

%A _Luciano Ancora_, Feb 05 2015

%E Edited by _Alonso del Arte_ and _Bruno Berselli_, Feb 10 2015

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)