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A382673 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^3.

%I A382673 #21 Apr 06 2025 03:48:13
%S A382673 1,1,1,1,4,1,1,10,10,1,1,22,52,22,1,1,46,208,208,46,1,1,94,736,1372,
%T A382673 736,94,1,1,190,2440,7516,7516,2440,190,1,1,382,7792,37012,60316,
%U A382673 37012,7792,382,1,1,766,24328,170668,418996,418996,170668,24328,766,1,1,1534,74896,754132,2653036,3964684,2653036,754132,74896,1534,1
%N A382673 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * k! * [x^n * y^k] exp(x+y) / (exp(x) + exp(y) - exp(x+y))^3.
%F A382673 E.g.f.: exp(x+y) / (exp(x) + exp(y) - exp(x+y))^3.
%F A382673 A(n,k) = A(k,n).
%F A382673 A(n,k) = Sum_{j=0..min(n,k)} (j!)^2 * binomial(j+2,2) * Stirling2(n+1,j+1) * Stirling2(k+1,j+1).
%e A382673 Square array begins:
%e A382673   1,  1,    1,     1,      1,       1, ...
%e A382673   1,  4,   10,    22,     46,      94, ...
%e A382673   1, 10,   52,   208,    736,    2440, ...
%e A382673   1, 22,  208,  1372,   7516,   37012, ...
%e A382673   1, 46,  736,  7516,  60316,  418996, ...
%e A382673   1, 94, 2440, 37012, 418996, 3964684, ...
%e A382673   ...
%o A382673 (PARI) a(n, k) = sum(j=0, min(n, k), j!^2*binomial(j+2, 2)*stirling(n+1, j+1, 2)*stirling(k+1, j+1, 2));
%Y A382673 Columns k=0..2 give A000012, A033484, A382675.
%Y A382673 Main diagonal gives A382676.
%Y A382673 Cf. A099594, A136126, A382674.
%Y A382673 Cf. A382735.
%K A382673 nonn,tabl
%O A382673 0,5
%A A382673 _Seiichi Manyama_, Apr 03 2025