%I #12 Oct 27 2024 10:16:24
%S 1,1,3,1,5,13,1,7,29,71,1,9,51,195,461,1,11,79,409,1493,3447,1,13,113,
%T 737,3623,12823,29093,1,15,153,1203,7427,35285,122125,273343,1,17,199,
%U 1831,13601,81009,375591,1277991,2829325,1,19,251,2645,22961,164371
%N Array read by antidiagonals: A(n,k) = (k+2)*A(n-1,k+1) + Sum_{j=0..k} A(n-1,j) with A(0,k) = 1, n >= 0, k >= 0.
%F Conjecture: A(n, 0) = A003319(n+2). - _Mikhail Kurkov_, Oct 27 2024
%e Array begins:
%e ===========================================================
%e n\k| 0 1 2 3 4 5 6 ...
%e ---+-------------------------------------------------------
%e 0 | 1 1 1 1 1 1 1 ...
%e 1 | 3 5 7 9 11 13 15 ...
%e 2 | 13 29 51 79 113 153 199 ...
%e 3 | 71 195 409 737 1203 1831 2645 ...
%e 4 | 461 1493 3623 7427 13601 22961 36443 ...
%e 5 | 3447 12823 35285 81009 164371 304667 526833 ...
%e 6 | 29093 122125 375591 954419 2124937 4289433 8025755 ...
%e ...
%o (PARI)
%o A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, 1)); r[1] = v[1..n+1];
%o for(i=1, m, v=vector(#v-1, k, (k+1)*v[k+1] + sum(j=1, k, v[j])); r[1+i] = v[1..n+1]); Mat(r)}
%o { A(6) }
%Y Row 2 appears to be essentially A144391. - _Joerg Arndt_, Feb 17 2024
%K nonn,tabl,changed
%O 0,3
%A _Mikhail Kurkov_, Feb 17 2024