cp's OEIS Frontend

A369527 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.

%I A369527 #14 Nov 24 2024 13:33:52
%S A369527 1,1,2,1,5,7,1,10,30,37,1,17,107,227,264,1,26,298,1261,2169,2433,1,37,
%T A369527 687,5455,16804,25480,27913,1,50,1382,18557,105837,257073,358993,
%U A369527 386906,1,65,2515,52267,516192,2209584,4523241,5959213,6346119,1,82,4242,127477,2009089,14913889,50267233,90976402,114813254,121159373
%N A369527 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 A369527 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 A369527 Array begins:
%e A369527 ====================================================
%e A369527 n\k|    0     1      2       3        4        5 ...
%e A369527 ---+------------------------------------------------
%e A369527 0  |    1     1      1       1        1        1 ...
%e A369527 1  |    2     5     10      17       26       37 ...
%e A369527 2  |    7    30    107     298      687     1382 ...
%e A369527 3  |   37   227   1261    5455    18557    52267 ...
%e A369527 4  |  264  2169  16804  105837   516192  2009089 ...
%e A369527 5  | 2433 25480 257073 2209584 14913889 78851808 ...
%e A369527   ...
%o A369527 (PARI)
%o A369527 A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, 1)); r[1] = v[1..n+1];
%o A369527 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 A369527 { A(5) }
%Y A369527 Column k=0 is A135920 (without initial term and with different offset).
%K A369527 nonn,tabl
%O A369527 0,3
%A A369527 _Mikhail Kurkov_, Jan 25 2024