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A258997 A(n,k) = pi-based antiderivative of n^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A258997 #34 Oct 26 2018 20:39:31
%S A258997 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 A258997 216,192,30,7,0,0,0,192,810,1024,225,84,4,0,0,0,448,2916,5120,1500,
%U A258997 756,56,12,0,0,0,1024,10206,24576,9375,6048,588,192,12,0
%N A258997 A(n,k) = pi-based antiderivative of n^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A258997 Alois P. Heinz, <a href="/A258997/b258997.txt">Antidiagonals n = 0..140, flattened</a>
%F A258997 A(n,k) = A258851(n^k) = k * n^(k-1) * A258851(n).
%e A258997 Square array A(n,k) begins:
%e A258997   0, 0,  0,   0,    0,     0,      0,       0, ...
%e A258997   0, 0,  0,   0,    0,     0,      0,       0, ...
%e A258997   0, 1,  4,  12,   32,    80,    192,     448, ...
%e A258997   0, 2, 12,  54,  216,   810,   2916,   10206, ...
%e A258997   0, 4, 32, 192, 1024,  5120,  24576,  114688, ...
%e A258997   0, 3, 30, 225, 1500,  9375,  56250,  328125, ...
%e A258997   0, 7, 84, 756, 6048, 45360, 326592, 2286144, ...
%e A258997   0, 4, 56, 588, 5488, 48020, 403368, 3294172, ...
%p A258997 with(numtheory):
%p A258997 d:= n-> n*add(i[2]*pi(i[1])/i[1], i=ifactors(n)[2]):
%p A258997 A:= (n, k)-> `if`(k=0, 0, k*n^(k-1)*d(n)):
%p A258997 seq(seq(A(n, h-n), n=0..h), h=0..14);
%Y A258997 Rows n=0+1,2,3,4,8 give: A000004, A001787, A212697, A018215, A230539.
%Y A258997 Columns k=0,1 give: A000004, A258851.
%Y A258997 Main diagonal gives A258846.
%Y A258997 Cf. A000720.
%K A258997 nonn,tabl
%O A258997 0,13
%A A258997 _Alois P. Heinz_, Jun 27 2015