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A143965 Factorial eigentriangle: A119502 * (A051295 *0^(n-k); 0<=k<=n 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

A119502) and the INVERT transform of the factorials (A051925) 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

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),... - Gary W. Adamson, Jul 21 2016

LINKS

Table of n, a(n) for n=0..44.

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.

EXAMPLE

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.

Sequence in context: A284466 A258615 A152431 * A182073 A098361 A050977

Adjacent sequences:  A143962 A143963 A143964 * A143966 A143967 A143968

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Sep 06 2008

STATUS

approved

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Last modified August 25 16:31 EDT 2019. Contains 326324 sequences. (Running on oeis4.)