A330151 Partial sums of 4th powers of the even numbers.
0, 16, 272, 1568, 5664, 15664, 36400, 74816, 140352, 245328, 405328, 639584, 971360, 1428336, 2042992, 2852992, 3901568, 5237904, 6917520, 9002656, 11562656, 14674352, 18422448, 22899904, 28208320, 34458320, 41769936, 50272992, 60107488, 71423984, 84383984
Offset: 0
Examples
a(4) = 0^4 + 2^4 + 4^4 + 6^4 + 8^4 = 5664.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Abdelkarim Assoul, The sum of the natural numbers peers, odd of p-th degree, hal-01924427, 2015.
- Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
Programs
-
Mathematica
a[n_] := (8/15)*n*(6*n^4 + 15*n^3 + 10*n^2 - 1); Array[a, 31, 0] (* Amiram Eldar, Dec 08 2019 *)
-
PARI
a(n) = sum(i=0, n, 16*i^4); \\ Jinyuan Wang, Dec 07 2019
-
PARI
concat(0, Vec(16*x*(1 + x)*(1 + 10*x + x^2) / (1 - x)^6 + O(x^30))) \\ Colin Barker, Dec 08 2019
-
Python
def A330151(n): return 8*n*(n**2*(n*(6*n + 15) + 10) - 1)//15 # Chai Wah Wu, Dec 07 2021
Formula
a(n) = Sum_{k=1..n} (2*k)^4 = (8/15)*n*(6*n^4 + 15*n^3 + 10*n^2 - 1).
a(n) = 16*A000538(n).
From Colin Barker, Dec 08 2019: (Start)
G.f.: 16*x*(1 + x)*(1 + 10*x + x^2) / (1 - x)^6.
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) for n>5.
(End)
E.g.f.: (8/15)*exp(x)*x*(30 + 225*x + 250*x^2 + 75*x^3 + 6*x^4). - Stefano Spezia, Dec 08 2019
Extensions
More terms from Jinyuan Wang, Dec 07 2019