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
Offset: 0
Examples
Triangle begins:
1;
1, 1;
2, 4, 2;
6, 18, 18, 6;
24, 96, 144, 96, 24;
120, 600, 1200, 1200, 600, 120;
...
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.
Crossrefs
Programs
-
Magma
/* As triangle */ [[Factorial(n)*Binomial(n, k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Sep 28 2015
-
Mathematica
Table[n!*Binomial[n, j], {n, 0, 30}, {j, 0, n}] (* G. C. Greubel, Sep 27 2015 *) -
Sage
factorial(n)*binomial(n,k) # Danny Rorabaugh, Sep 27 2015
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
Extensions
Name exchanged with a formula by Peter Luschny, Feb 01 2015
Comments