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

0,2

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).

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

Example

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

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 *)

Prog

(Magma) [(8/3)*n^2*(n+1)^2*(2*n^2+2*n-1): n in [0..30]]; // 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

Crossrefs

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

Sequence in context: A144319 A041481 A158685 * A159360 A249392 A114933

Adjacent sequences: A265018 A265019 A265020 * A265022 A265023 A265024

Keyword

nonn

Author

Assoul Abdelkarim, Nov 30 2015

Status

approved