A196347 - OEIS

A196347

Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).

1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 24, 96, 144, 96, 24, 120, 600, 1200, 1200, 600, 120, 720, 4320, 10800, 14400, 10800, 4320, 720, 5040, 35280, 105840, 176400, 176400, 105840, 35280, 5040, 40320, 322560, 1128960, 2257920, 2822400, 2257920, 1128960, 322560, 40320

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OFFSET

0,4

COMMENTS

Unsigned version of A021012.

Equal to A136572*A007318.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..495

P. Bala, Deformations of the Hadamard product of power series

Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.

M. Dukes, C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

FORMULA

T(n,k) is given by (1,1,2,2,3,3,4,4,5,5,6,6,...) DELTA (1,1,2,2,3,3,4,4,5,5,6,6, ...) where DELTA is the operator defined in A084938.

Sum_{k>=0} T(m,k)*T(n,k) = (m+n)!.

T(2n,n) = A122747(n).

Sum_{k>=0} T(n,k)^2 = A010050(n) = (2n)!.

Sum_{k>=0} T(n,k)*x^k = A000007(n), A000142(n), A000165(n), A032031(n), A047053(n), A052562(n), A047058(n), A051188(n), A051189(n), A051232(n), A051262(n), A196258(n), A145448(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.

The row polynomials have the form (x + 1) o (x + 2) o ... o (x + n), where o denotes the black diamond multiplication operator of Dukes and White. See example E10 in the Bala link. - Peter Bala, Jan 18 2018

EXAMPLE

Triangle begins:

1, 1;

2, 4, 2;

6, 18, 18, 6;

24, 96, 144, 96, 24;

120, 600, 1200, 1200, 600, 120;

...

MATHEMATICA

Table[n!*Binomial[n, j], {n, 0, 30}, {j, 0, n}] (* G. C. Greubel, Sep 27 2015 *)

PROG

(Sage) factorial(n)*binomial(n, k) # Danny Rorabaugh, Sep 27 2015