A143965 - OEIS

A143965

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

%I #22 Jul 24 2024 09:48:18

%S 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,

%T 54,235,5040,720,240,120,90,108,235,1237,40320,5040,1440,600,360,324,

%U 470,1237,7790

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

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

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

%C (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; ...).

%C 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.

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

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

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

%C 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

%C From _Gary W. Adamson_, Jul 21 2016: (Start)

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

%C 1, 1, 0, 0, 0, 0, ...

%C 1, 0, 2, 0, 0, 0, ...

%C 1, 0, 0, 3, 0, 0, ...

%C 1, 0, 0, 0, 4, 0, ...

%C 1, 0, 0, 0, 0, 5, ...

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

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

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

%e First few rows of the triangle:

%e 1;

%e 1, 1;

%e 2, 1, 2;

%e 6, 2, 2, 5;

%e 24, 6, 4, 5, 15;

%e 120, 24, 12, 10, 15, 54;

%e 720, 120, 48, 30, 30, 54, 235;

%e 5040, 720, 240, 120, 90, 108, 235, 1737;

%e ...

%e 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).

%Y Cf. A000142, A051295, A119502.

%K nonn,tabl

%O 0,4

%A _Gary W. Adamson_, Sep 06 2008