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A193561 Augmentation of the triangle A004736. See Comments.

%I A193561 #5 Mar 30 2012 18:57:38
%S A193561 1,2,1,6,6,3,24,36,30,15,120,240,270,210,105,720,1800,2520,2520,1890,
%T A193561 945,5040,15120,25200,30240,28350,20790,10395,40320,141120,272160,
%U A193561 378000,415800,374220,270270,135135,362880,1451520,3175200,4989600
%N A193561 Augmentation of the triangle A004736.  See Comments.
%C A193561 For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
%C A193561 Regarding A193561, if the triangle is written as (w(n,k)), then
%C A193561 w(n,n)=A001147(n), "double factorial numbers";
%C A193561 w(n,n-1)=A097801(n), (2n)!/(n!*2^(n-1))
%C A193561 col 1:  A000142, n!
%C A193561 col 2: A001286, Lah numbers, (n-1)*n!/2
%e A193561 First 5 rows of A193560:
%e A193561 1
%e A193561 2.....1
%e A193561 6.....6....3
%e A193561 24....36...30...15
%e A193561 120...240..270..210..105
%t A193561 p[n_, k_] := n + 1 - k
%t A193561 Table[p[n, k], {n, 0, 5}, {k, 0, n}]  (* A004736 *)
%t A193561 m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
%t A193561 TableForm[m[4]]
%t A193561 w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
%t A193561 v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
%t A193561 v[n_] := v[n - 1].m[n]
%t A193561 TableForm[Table[v[n], {n, 0, 6}]]  (* A193561 *)
%t A193561 Flatten[Table[v[n], {n, 0, 8}]]
%Y A193561 Cf. A193091.
%K A193561 nonn,tabl
%O A193561 0,2
%A A193561 _Clark Kimberling_, Jul 30 2011