A062275 - OEIS

A062275

Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

%I #29 Jun 14 2018 18:49:12

%S 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,

%T 800,5184,5184,800,6,0,0,7,2304,30375,65536,30375,2304,7,0,0,8,6272,

%U 157464,640000,640000,157464,6272,8,0,0,9,16384,750141,5308416,9765625

%N Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

%C Here 0^0 is defined to be 1. - _Wolfdieter Lang_, May 27 2018

%H Eric Chen, <a href="/A062275/b062275.txt">Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)</a>

%F From _Wolfdieter Lang_, May 22 2018: (Start)

%F As a sequence: a(n) = A003992(n)*A004248(n).

%F As a triangle: T(n, k) = (n-k)^k * k^(n-k), for n >= 1 and k = 1..n. (End)

%e A(3, 2) = 3^2 * 2^3 = 9*8 = 72.

%e The array A(n, k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 10 ...

%e 0: 1 0 0 0 0 0 0 0 0 0 0 ...

%e 1: 0 1 2 3 4 5 6 7 8 9 10 ...

%e 2: 0 2 16 72 256 800 2304 6272 16384 41472 102400 ...

%e 3: 0 3 72 729 5184 30375 157464 750141 3359232 14348907 59049000 ...

%e ...

%e The triangle T(n, k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 ...

%e 0: 1

%e 1: 0 0

%e 2: 0 1 0

%e 3: 0 2 2 0

%e 4: 0 3 16 3 0

%e 5: 0 4 72 72 4 0

%e 6: 0 5 256 729 256 5 0

%e 7: 0 6 800 5184 5184 800 6 0

%e 8: 0 7 2304 30375 65536 30375 2304 7 0

%e 9: 0 8 6272 157464 640000 640000 157464 6272 8 0

%e ... - _Wolfdieter Lang_, May 22 2018

%t {{1}}~Join~Table[(#^k k^#) &[n - k], {n, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, May 24 2018 *)

%o (PARI) t1(n)=n-binomial(round(sqrt(2+2*n)), 2)

%o t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)

%o a(n)=t1(n)^t2(n)*t2(n)^t1(n) \\ _Eric Chen_, Jun 09 2018

%Y 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

%Y Cf. A055651, A055652, A303990.

%K nonn,tabl,easy

%O 0,8

%A _Henry Bottomley_, Jul 02 2001