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A383843 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/Product_{j=0..k} (1 - j*x)^2.

%I A383843 #22 May 15 2025 08:17:09
%S A383843 1,1,0,1,2,0,1,6,3,0,1,12,23,4,0,1,20,86,72,5,0,1,30,230,480,201,6,0,
%T A383843 1,42,505,2000,2307,522,7,0,1,56,973,6300,14627,10044,1291,8,0,1,72,
%U A383843 1708,16464,65002,95060,40792,3084,9,0,1,90,2796,37632,227542,587580,567240,157440,7181,10,0
%N A383843 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/Product_{j=0..k} (1 - j*x)^2.
%F A383843 A(n,k) = Sum_{j=0..n} Stirling2(j+k,k) * Stirling2(n-j+k,k).
%e A383843 Square array begins:
%e A383843   1, 1,    1,     1,      1,       1,        1, ...
%e A383843   0, 2,    6,    12,     20,      30,       42, ...
%e A383843   0, 3,   23,    86,    230,     505,      973, ...
%e A383843   0, 4,   72,   480,   2000,    6300,    16464, ...
%e A383843   0, 5,  201,  2307,  14627,   65002,   227542, ...
%e A383843   0, 6,  522, 10044,  95060,  587580,  2725380, ...
%e A383843   0, 7, 1291, 40792, 567240, 4817990, 29331038, ...
%o A383843 (PARI) a(n, k) = sum(j=0, n, stirling(j+k, k, 2)*stirling(n-j+k, k, 2));
%Y A383843 Columns k=0..4 give A000007, A000027(n+1), A045618, A383841, A383842.
%Y A383843 Main diagonal gives A350376.
%Y A383843 A(n,n-1) gives A383880.
%Y A383843 Cf. A106800, A287532.
%K A383843 nonn,tabl
%O A383843 0,5
%A A383843 _Seiichi Manyama_, May 12 2025