A253712 - OEIS

%I #36 Sep 08 2022 08:46:10

%S 1,4098,539636,17852390,279305769,2717541484,18997064400,103996064052,

%T 471424600185,1838853136318,6344710049172,19766667410282,

%U 56486709893873,149900664752760,373060957502272,877696226962440,1964953733652369,4209042621768474,8666446428950740,17219850236133006,33129081554701913,61893315504320036

%N Second partial sums of 12th powers (A008456).

%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 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+1)^2*n*(n+2)*(30*n^10+300*n^9+925*n^8+200*n^7-3022*n^6-772*n^5+7073*n^4-1228*n^3-7888*n^2+5528*n-691)/5460.

%F a(n) = 2*a(n-1)-a(n-2)+n^12.

%F G.f.: x*(1 + 4083*x + 478271*x^2 + 10187685*x^3 + 66318474*x^4 + 162512286*x^5 + 162512286*x^6 + 66318474*x^7 + 10187685*x^8 + 478271*x^9 + 4083*x^10 + x^11)/(1-x)^15. - _Vincenzo Librandi_, Jan 19 2015

%t RecurrenceTable[{a[n] == 2 a[n - 1] - a[n - 2] + n^12, a[1] == 1, a[2] == 4098}, a, {n, 1, 25}] (* _Bruno Berselli_, Jan 19 2015 *)

%t Table[(n + 1)^2 n (n + 2) (30 n^10 + 300 n^9 + 925 n^8 + 200 n^7 - 3022 n^6 - 772 n^5 + 7073 n^4 - 1228 n^3 - 7888 n^2 + 5528 n - 691)/5460, {n, 1, 25}] (* _Vincenzo Librandi_, Jan 19 2015 *)

%t Nest[Accumulate[#]&,Range[30]^12,2] (* _Harvey P. Dale_, Aug 17 2020 *)

%o (Magma) [(n+1)^2*n*(n+2)*(30*n^10+300*n^9+925*n^8+200*n^7-3022*n^6-772*n^5+7073*n^4-1228*n^3-7888*n^2+5528*n-691)/5460: n in [1..30]]; // _Vincenzo Librandi_, Jan 19 2015

%K nonn

%O 1,2

%A _Luciano Ancora_, Jan 12 2015

%E a(22) corrected by _Vincenzo Librandi_, Jan 19 2015