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A193561 Augmentation of the triangle A004736.  See Comments. 2

%I

%S 1,2,1,6,6,3,24,36,30,15,120,240,270,210,105,720,1800,2520,2520,1890,

%T 945,5040,15120,25200,30240,28350,20790,10395,40320,141120,272160,

%U 378000,415800,374220,270270,135135,362880,1451520,3175200,4989600

%N Augmentation of the triangle A004736. See Comments.

%C For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

%C Regarding A193561, if the triangle is written as (w(n,k)), then

%C w(n,n)=A001147(n), "double factorial numbers";

%C w(n,n-1)=A097801(n), (2n)!/(n!*2^(n-1))

%C col 1: A000142, n!

%C col 2: A001286, Lah numbers, (n-1)*n!/2

%e First 5 rows of A193560:

%e 1

%e 2.....1

%e 6.....6....3

%e 24....36...30...15

%e 120...240..270..210..105

%t p[n_, k_] := n + 1 - k

%t Table[p[n, k], {n, 0, 5}, {k, 0, n}] (* A004736 *)

%t m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]

%t TableForm[m[4]]

%t w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];

%t v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};

%t v[n_] := v[n - 1].m[n]

%t TableForm[Table[v[n], {n, 0, 6}]] (* A193561 *)

%t Flatten[Table[v[n], {n, 0, 8}]]

%Y Cf. A193091.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Jul 30 2011

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Last modified March 26 10:18 EDT 2019. Contains 321491 sequences. (Running on oeis4.)