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A256892 Triangular array read by rows, the matrix product of the unsigned Lah numbers and the Stirling set numbers, T(n,k) for n>=0 and 0<=k<=n.

%I A256892 #21 Jul 16 2025 20:41:19
%S A256892 1,0,1,0,3,1,0,13,9,1,0,73,79,18,1,0,501,755,265,30,1,0,4051,7981,
%T A256892 3840,665,45,1,0,37633,93135,57631,13580,1400,63,1,0,394353,1192591,
%U A256892 911582,274141,38290,2618,84,1,0,4596553,16645431,15285313,5633922,999831,92358,4494,108,1
%N A256892 Triangular array read by rows, the matrix product of the unsigned Lah numbers and the Stirling set numbers, T(n,k) for n>=0 and 0<=k<=n.
%C A256892 Also the Bell transform of A000262(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016
%F A256892 T(n+1,1) = A000262(n).
%F A256892 T(n+1,n) = A045943(n).
%F A256892 Row sums are A084357.
%e A256892 Triangle starts:
%e A256892 1;
%e A256892 0,    1;
%e A256892 0,    3,    1;
%e A256892 0,   13,    9,    1;
%e A256892 0,   73,   79,   18,   1;
%e A256892 0,  501,  755,  265,  30,  1;
%e A256892 0, 4051, 7981, 3840, 665, 45, 1;
%p A256892 # The function BellMatrix is defined in A264428.
%p A256892 BellMatrix(n -> simplify(hypergeom([-n, -n-1], [], 1)), 9); # _Peter Luschny_, Jan 29 2016
%t A256892 BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A256892 B = BellMatrix[Function[n, HypergeometricPFQ[{-n, -n-1}, {}, 1]], rows = 12];
%t A256892 Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%o A256892 (SageMath)
%o A256892 def Lah(n, k):
%o A256892     if n == k: return 1
%o A256892     if k<0 or  k>n: return 0
%o A256892     return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
%o A256892 matrix(ZZ, 8, Lah) * matrix(ZZ, 8, stirling_number2)  # as a square matrix
%Y A256892 See also A088814 and A088729 for variants based on an (1,1)-offset of the number triangles. See A131222 for the product Lah * Stirling-cycle.
%Y A256892 A079640 is an unsigned matrix inverse reduced to an (1,1)-offset.
%Y A256892 Cf. A000262, A045943, A084357, A088729, A088814.
%K A256892 nonn,tabl,easy
%O A256892 0,5
%A A256892 _Peter Luschny_, Apr 12 2015