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%I #34 Sep 08 2022 08:46:10
%S 1,8194,1610710,70322090,1359736595,15709845116,126948964044,
%T 787943896860,3990804658005,17193665419150,64919238324226,
%U 219638016608374,677231901484775,1928540559615320,5126044286105240,12827147639965656,30432829026732009,68861475279169530,149343104993864110,311744734708558690,628618742162372731
%N Second partial sums of 13th powers (A010801).
%C The formula for the second partial sums of m-th powers is: b(n,m) = (n+1)*F(m) - F(m+1), where F(m) are the m-th Faulhaber's formulas.
%H Harvey P. Dale, <a href="/A253713/b253713.txt">Table of n, a(n) for n = 1..1000</a>
%H Luciano Ancora, <a href="/A253636/a253636_3.pdf">Recurrence relation for the second partial sums of m-th powers</a>
%H Luciano Ancora, <a href="/A253636/a253636_4.pdf">Second partial sums of the m-th powers</a>
%F a(n) = n*(n+1)*(n+2)*(6*n^12+72*n^11+297*n^10+330*n^9-765*n^8-1368*n^7+2059*n^6+2994*n^5-4091*n^4-2724*n^3+4069*n^2+66*n-735)/1260.
%F a(n) = 2*a(n-1)-a(n-2)+n^13.
%t Table[n (n+1) (n+2) (6 n^12 + 72 n^11 + 297 n^10 + 330 n^9 - 765 n^8 - 1368 n^7 + 2059 n^6 + 2994 n^5 - 4091 n^4 -2724 n^3 + 4069 n^2 + 66 n - 735) / 1260, {n, 40}] (* _Vincenzo Librandi_, Jan 19 2015 *)
%t Nest[Accumulate,Range[30]^13,2] (* _Harvey P. Dale_, Jul 24 2018 *)
%o (Magma) [n*(n+1)*(n+2)*(6*n^12+72*n^11+297*n^10+330*n^9-765*n^8-1368*n^7+2059*n^6+2994*n^5-4091*n^4-2724*n^3+4069*n^2+66*n-735)/1260: n in [1..30]]; // _Vincenzo Librandi_, Jan 19 2015
%K nonn,easy
%O 1,2
%A _Luciano Ancora_, Jan 12 2015
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