%I #44 Sep 08 2022 08:45:59
%S 1,1,1,2,4,2,6,18,18,6,24,96,144,96,24,120,600,1200,1200,600,120,720,
%T 4320,10800,14400,10800,4320,720,5040,35280,105840,176400,176400,
%U 105840,35280,5040,40320,322560,1128960,2257920,2822400,2257920,1128960,322560,40320
%N Triangle T(n, k) read by rows, T(n, k) = n!*binomial(n, k).
%C Unsigned version of A021012.
%C Equal to A136572*A007318.
%H G. C. Greubel, <a href="/A196347/b196347.txt">Table of n, a(n) for n = 0..495</a>
%H P. Bala, <a href="/A131689/a131689.pdf">Deformations of the Hadamard product of power series</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2101.06713">On the inversion of Riordan arrays</a>, arXiv:2101.06713 [math.CO], 2021.
%H M. Dukes, C. D. White, <a href="http://arxiv.org/abs/1603.01589">Web Matrices: Structural Properties and Generating Combinatorial Identities</a>, arXiv:1603.01589 [math.CO], 2016.
%F 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.
%F Sum_{k>=0} T(m,k)*T(n,k) = (m+n)!.
%F T(2n,n) = A122747(n).
%F Sum_{k>=0} T(n,k)^2 = A010050(n) = (2n)!.
%F 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.
%F 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
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 4, 2;
%e 6, 18, 18, 6;
%e 24, 96, 144, 96, 24;
%e 120, 600, 1200, 1200, 600, 120;
%e ...
%t Table[n!*Binomial[n, j], {n, 0, 30}, {j, 0, n}] (* _G. C. Greubel_, Sep 27 2015 *)
%o (Sage) factorial(n)*binomial(n,k) # _Danny Rorabaugh_, Sep 27 2015
%o (Magma) /* As triangle */ [[Factorial(n)*Binomial(n, k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Sep 28 2015
%Y Cf. A007318, A008619, A021012, A084938, A136572.
%Y Columns include: A000142, A001563, A001804, A001805, A001806, A001807.
%K nonn,tabl,easy
%O 0,4
%A _Philippe Deléham_, Oct 28 2011
%E Name exchanged with a formula by _Peter Luschny_, Feb 01 2015