A265021 Sum of fifth powers of the first n even numbers.

0, 32, 1056, 8832, 41600, 141600, 390432, 928256, 1976832, 3866400, 7066400, 12220032, 20182656, 32064032, 49274400, 73574400, 107128832, 152564256, 213030432, 292265600, 394665600, 525356832, 690273056, 896236032, 1151040000, 1463540000, 1843744032

Offset: 0

Author

Assoul Abdelkarim, Nov 30 2015

Keywords

• nonn

Examples

			a(4) =  2^5 + 4^5 + 6^5 + 8^5 = 41600.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Assoul Abdelkarim, The sums of powers of integers natural even, odd, RMSS-D-15-00105.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A000539, A002594 (the same for odd numbers).

Programs

Magma

[(8/3)*n^2*(n+1)^2*(2*n^2+2*n-1): n in [0..30]]; // Vincenzo Librandi, Dec 01 2015

Mathematica

Accumulate[Range[0, 60, 2]^5] (* Michael De Vlieger, Nov 30 2015 *)
CoefficientList[Series[32 x (1 + 26 x + 66 x^2 + 26 x^3 + x^4)/(1 - x)^7, {x, 0, 33}], x] (* Vincenzo Librandi, Dec 01 2015 *)

PARI

vector(100, n, n--; (8/3)*n^2*(n+1)^2*(2*n^2+2*n-1)) \\ Altug Alkan, Dec 01 2015

Formula

a(n) = 32 * Sum_{i=0..n} i^5 = (8/3)*n^2*(n+1)^2*(2*n^2+2*n-1).

a(n) = 32 * A000539(n).

G.f.: 32*x*(1 + 26*x + 66*x^2 + 26*x^3 + x^4)/(1-x)^7. - Vincenzo Librandi, Dec 01 2015

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). - Vincenzo Librandi, Dec 01 2015