A198064 Triangle read by rows (n >= 0, 0 <= k <= n, m = 4); T(n,k) = Sum{j=0..m} Sum{i=0..m} (-1)^(j+i)*C(i,j)*n^j*k^(m-j).

0, 1, 1, 16, 5, 16, 81, 31, 31, 81, 256, 121, 80, 121, 256, 625, 341, 211, 211, 341, 625, 1296, 781, 496, 405, 496, 781, 1296, 2401, 1555, 1031, 781, 781, 1031, 1555, 2401, 4096, 2801, 1936, 1441, 1280, 1441, 1936, 2801, 4096, 6561, 4681, 3355, 2511, 2101

Offset

0,4

Links

Table of n, a(n) for n=0..49.

Formula

T(n,k) = k^4-2*k^3*n+4*k^2*n^2-3*k*n^3+n^4.

T(n,0) = T(n,n) = n^m = n^4 = A000583(n).

T(2n,n) = (m+1)n^m = 5n^4.

T(2n+1,n+1) = (n+1)^(m+1)-n^(m+1) = (n+1)^5-n^5 = A022521(n).

Sum{k=0..n} T(n,k) = (16n^5+30n^4+15n^3-n)/30.

T(n+1,k+1)C(n,k)^5/(k+1)^4 = A197654(n,k).

Example

[0]                      0
[1]                     1, 1
[2]                  16, 5, 16
[3]                81, 31, 31, 81
[4]            256, 121, 80, 121, 256
[5]         625, 341, 211, 211, 341, 625
[6]     1296, 781, 496, 405, 496, 781, 1296
[7] 2401, 1555, 1031, 781, 781, 1031, 1555, 2401

Maple

A198064 := (n, k) -> k^4-2*k^3*n+4*k^2*n^2-3*k*n^3+n^4:

Crossrefs

Cf. A057427, A003056, A073254, A198063, A198065.

Sequence in context: A040245 A070538 A296336 * A233000 A070581 A057995

Adjacent sequences: A198061 A198062 A198063 * A198065 A198066 A198067

Keyword

nonn,tabl

Author

Peter Luschny, Oct 26 2011

Status

approved