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%I #14 Nov 24 2024 13:33:52
%S 1,1,2,1,5,7,1,10,30,37,1,17,107,227,264,1,26,298,1261,2169,2433,1,37,
%T 687,5455,16804,25480,27913,1,50,1382,18557,105837,257073,358993,
%U 386906,1,65,2515,52267,516192,2209584,4523241,5959213,6346119,1,82,4242,127477,2009089,14913889,50267233,90976402,114813254,121159373
%N Array read by downward antidiagonals: A(n,k) = (k+1)^2*A(n-1,k) + A(n-1,k+1) with A(0,k) = 1, n >= 0, k >= 0.
%H Ira M. Gessel, <a href="https://mathoverflow.net/a/462813/231922">General case of the some R-recursions</a>, answer to question on MathOverflow (2024).
%e Array begins:
%e ====================================================
%e n\k| 0 1 2 3 4 5 ...
%e ---+------------------------------------------------
%e 0 | 1 1 1 1 1 1 ...
%e 1 | 2 5 10 17 26 37 ...
%e 2 | 7 30 107 298 687 1382 ...
%e 3 | 37 227 1261 5455 18557 52267 ...
%e 4 | 264 2169 16804 105837 516192 2009089 ...
%e 5 | 2433 25480 257073 2209584 14913889 78851808 ...
%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^2*v[k] + v[k+1]); r[1+i] = v[1..n+1]); Mat(r)}
%o { A(5) }
%Y Column k=0 is A135920 (without initial term and with different offset).
%K nonn,tabl
%O 0,3
%A _Mikhail Kurkov_, Jan 25 2024