A143965 Factorial eigentriangle: A119502 * (A051295 *0^(n-k)); 0 <= k <= n.

1, 1, 1, 2, 1, 2, 6, 2, 2, 5, 24, 6, 4, 5, 15, 120, 24, 12, 10, 15, 54, 720, 120, 48, 30, 30, 54, 235, 5040, 720, 240, 120, 90, 108, 235, 1237, 40320, 5040, 1440, 600, 360, 324, 470, 1237, 7790

Offset: 0

Author

Gary W. Adamson, Sep 06 2008

Keywords

• nonn
• tabl

Comments

Triangle read by rows, termwise product of (n-k)! (i.e factorial decrescendo,

A119502) and the INVERT transform of the factorials (A051295) prefaced by a 1:

(1, 1, 2, 5, 15, 54, 235, 1237, 7790, ...). A119502 = (1; 1,1; 2,1,1; 6,2,1,1; 24,6,2,1,1; ...).

The operation (A051295 * 0^(n-k)) with A051295 prefaced with a 1 = an infinite lower triangular matrix with (1, 1, 2, 5, 15, 54, 235, ...) in the main diagonal and the rest zeros.

Row sums = the INVERT transform of the factorials, A051295: (1, 2, 5, 15, 54, 235, 1237, ...).

Right border shifts A051295: (1, 1, 2, 5, 15, ...).

Sum of n-th row terms = rightmost term of next row; e.g. ( 6 + 2 + 2 + 5) = 15.

With offset 1 for n and k, T(n,k) counts permutations of [n] that contain a 132 pattern only as part of a 4132 pattern by position k of largest entry n. Example: T(5,3)=4 counts 34512, 34521, 43512, 43521. - David Callan, Nov 21 2011

From Gary W. Adamson, Jul 21 2016: (Start)

A production matrix M for the reversal of the triangle is follows: M =

1, 1, 0, 0, 0, 0, ...

1, 0, 2, 0, 0, 0, ...

1, 0, 0, 3, 0, 0, ...

1, 0, 0, 0, 4, 0, ...

1, 0, 0, 0, 0, 5, ...

... Take powers of M, extracting the top row, getting: (1), (1, 1), (2, 1, 2), (5, 2, 2, 6), ... (End)

Examples

			First few rows of the triangle:
     1;
     1,   1;
     2,   1,   2;
     6,   2,   2,   5;
    24,   6,   4,   5, 15;
   120,  24,  12,  10, 15,  54;
   720, 120,  48,  30, 30,  54, 235;
  5040, 720, 240, 120, 90, 108, 235, 1737;
  ...
Example: Row 3 = (6, 2, 2, 5) = termwise products of row 3 terms of triangle A119502 (6, 2, 1, 1) and the first four terms of (1, 1, 2, 5, ...) = (6*1, 2*1, 1*2, 1*5).
		

Crossrefs

Cf. A000142, A051295, A119502.

Formula

Factorial eigentriangle: A119502 * (A051295 *0^(n-k)); 0 <= k <= n.

The operation uses A119502 prefaced with a 1 = (1, 1, 2, 5, 15, 54, 235, ...); i.e., the right border of the triangle.