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A379458 Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + 2*(k+1)!*Sum_{j=0..k} A(n-1,j)/j! with A(0,k) = 1, n >= 0, k >= 0.

%I A379458 #5 Jan 03 2025 02:03:11
%S A379458 1,1,3,1,9,15,1,31,79,109,1,129,459,835,1053,1,651,3003,6885,10661,
%T A379458 12767,1,3913,22183,61735,114373,161229,186763,1,27399,183975,603565,
%U A379458 1307997,2134803,2830787,3204313,1,219201,1698819,6424059,15981869,29753069,44649839,56720039,63128665
%N A379458 Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + 2*(k+1)!*Sum_{j=0..k} A(n-1,j)/j! with A(0,k) = 1, n >= 0, k >= 0.
%F A379458 Conjecture: A(n,0) = A217061(n+1).
%e A379458 Array begins:
%e A379458 ===========================================================
%e A379458 n\k|     0      1       2        3         4          5 ...
%e A379458 ---+-------------------------------------------------------
%e A379458 0  |     1      1       1        1         1          1 ...
%e A379458 1  |     3      9      31      129       651       3913 ...
%e A379458 2  |    15     79     459     3003     22183     183975 ...
%e A379458 3  |   109    835    6885    61735    603565    6424059 ...
%e A379458 4  |  1053  10661  114373  1307997  15981869  208612693 ...
%e A379458 5  | 12767 161229 2134803 29753069 437287383 6780218397 ...
%e A379458   ...
%o A379458 (PARI) A(m, n=m)={my(r=vectorv(m+1), v=vector(n+m+1, k, 1)); r[1] = v[1..n+1];
%o A379458 for(i=1, m, v=vector(#v-1, k, v[k+1] + 2*k!*sum(j=1, k, v[j]/(j-1)!)); r[1+i] = v[1..n+1]); Mat(r)}
%o A379458 { A(5) }
%Y A379458 Cf. A217061.
%K A379458 nonn,tabl
%O A379458 0,3
%A A379458 _Mikhail Kurkov_, Dec 23 2024