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A062275
Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.
4
1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 16, 3, 0, 0, 4, 72, 72, 4, 0, 0, 5, 256, 729, 256, 5, 0, 0, 6, 800, 5184, 5184, 800, 6, 0, 0, 7, 2304, 30375, 65536, 30375, 2304, 7, 0, 0, 8, 6272, 157464, 640000, 640000, 157464, 6272, 8, 0, 0, 9, 16384, 750141, 5308416, 9765625
OFFSET
0,8
COMMENTS
Here 0^0 is defined to be 1. - Wolfdieter Lang, May 27 2018
FORMULA
From Wolfdieter Lang, May 22 2018: (Start)
As a sequence: a(n) = A003992(n)*A004248(n).
As a triangle: T(n, k) = (n-k)^k * k^(n-k), for n >= 1 and k = 1..n. (End)
EXAMPLE
A(3, 2) = 3^2 * 2^3 = 9*8 = 72.
The array A(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1 0 0 0 0 0 0 0 0 0 0 ...
1: 0 1 2 3 4 5 6 7 8 9 10 ...
2: 0 2 16 72 256 800 2304 6272 16384 41472 102400 ...
3: 0 3 72 729 5184 30375 157464 750141 3359232 14348907 59049000 ...
...
The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 ...
0: 1
1: 0 0
2: 0 1 0
3: 0 2 2 0
4: 0 3 16 3 0
5: 0 4 72 72 4 0
6: 0 5 256 729 256 5 0
7: 0 6 800 5184 5184 800 6 0
8: 0 7 2304 30375 65536 30375 2304 7 0
9: 0 8 6272 157464 640000 640000 157464 6272 8 0
... - Wolfdieter Lang, May 22 2018
MATHEMATICA
{{1}}~Join~Table[(#^k k^#) &[n - k], {n, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 24 2018 *)
PROG
(PARI) t1(n)=n-binomial(round(sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
a(n)=t1(n)^t2(n)*t2(n)^t1(n) \\ Eric Chen, Jun 09 2018
CROSSREFS
Columns and rows of A, or columns and diagonals of T, include A000007, A001477, A007758, A062074, A062075 etc. Diagonals of A include A062206, A051443, A051490. Sum of rows of T are A062817(n), for n >= 1
Sequence in context: A355664 A254040 A376725 * A138270 A317643 A179011
KEYWORD
nonn,tabl,easy
AUTHOR
Henry Bottomley, Jul 02 2001
STATUS
approved