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A325146 A(n, k) = Stirling2(n + k, k)*A053657(n)*k!/(n + k)!, array read by ascending antidiagonals for n >= 0 and k >= 0.

%I A325146 #16 Jan 05 2024 13:30:06
%S A325146 1,0,1,0,1,1,0,4,2,1,0,2,14,3,1,0,48,12,30,4,1,0,16,496,36,52,5,1,0,
%T A325146 576,288,2064,80,80,6,1,0,144,18288,1656,5832,150,114,7,1,0,3840,8160,
%U A325146 145200,5920,13240,252,154,8,1
%N A325146 A(n, k) = Stirling2(n + k, k)*A053657(n)*k!/(n + k)!, array read by ascending antidiagonals for n >= 0 and k >= 0.
%e A325146 [0] 1,   1,     1,      1,      1,       1,       1,        1, ... A000012
%e A325146 [1] 0,   1,     2,      3,      4,       5,       6,        7, ... A001477
%e A325146 [2] 0,   4,    14,     30,     52,      80,     114,      154, ... A049451
%e A325146 [3] 0,   2,    12,     36,     80,     150,     252,      392, ... A011379
%e A325146 [4] 0,  48,   496,   2064,   5832,   13240,   26088,    46536, ...
%e A325146 [5] 0,  16,   288,   1656,   5920,   16200,   37296,    76048, ...
%e A325146 [6] 0, 576, 18288, 145200, 654816, 2153280, 5775936, 13429248, ...
%e A325146      A163176
%p A325146 A := (n, k) -> Stirling2(n + k, k)*A053657(n)*k!/(n + k)!:
%p A325146 seq(seq(A(n - k, k), k=0..n), n=0..10);
%t A325146 a053657[n_] := Product[p^Sum[Floor[(n-1) / ((p-1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}];
%t A325146 A[n_, k_] := StirlingS2[n+k, k] a053657[n+1] k! / (n+k)!;
%t A325146 Table[A[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 21 2019 *)
%Y A325146 Rows include A001477, A049451, A011379. Columns include A163176.
%Y A325146 Cf. A053657.
%K A325146 nonn,tabl
%O A325146 0,8
%A A325146 _Peter Luschny_, May 22 2019