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A(n,k) = pi-based antiderivative of n^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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%I #34 Oct 26 2018 20:39:31

%S 0,0,0,0,0,0,0,0,1,0,0,0,4,2,0,0,0,12,12,4,0,0,0,32,54,32,3,0,0,0,80,

%T 216,192,30,7,0,0,0,192,810,1024,225,84,4,0,0,0,448,2916,5120,1500,

%U 756,56,12,0,0,0,1024,10206,24576,9375,6048,588,192,12,0

%N A(n,k) = pi-based antiderivative of n^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%H Alois P. Heinz, <a href="/A258997/b258997.txt">Antidiagonals n = 0..140, flattened</a>

%F A(n,k) = A258851(n^k) = k * n^(k-1) * A258851(n).

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

%e 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 1, 4, 12, 32, 80, 192, 448, ...

%e 0, 2, 12, 54, 216, 810, 2916, 10206, ...

%e 0, 4, 32, 192, 1024, 5120, 24576, 114688, ...

%e 0, 3, 30, 225, 1500, 9375, 56250, 328125, ...

%e 0, 7, 84, 756, 6048, 45360, 326592, 2286144, ...

%e 0, 4, 56, 588, 5488, 48020, 403368, 3294172, ...

%p with(numtheory):

%p d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):

%p A:= (n, k)-> `if`(k=0, 0, k*n^(k-1)*d(n)):

%p seq(seq(A(n, h-n), n=0..h), h=0..14);

%Y Rows n=0+1,2,3,4,8 give: A000004, A001787, A212697, A018215, A230539.

%Y Columns k=0,1 give: A000004, A258851.

%Y Main diagonal gives A258846.

%Y Cf. A000720.

%K nonn,tabl

%O 0,13

%A _Alois P. Heinz_, Jun 27 2015