A113129 - OEIS

%I #11 Sep 08 2013 13:30:49

%S 1,0,1,0,0,2,0,0,1,6,0,0,0,10,24,0,0,0,4,82,120,0,0,0,0,84,672,720,0,

%T 0,0,0,27,1236,5820,5040,0,0,0,0,0,930,16328,54288,40320,0,0,0,0,0,

%U 248,20850,211080,548496,362880,0,0,0,0,0,0,12452,396528,2775432

%N Triangle T(n,k), 0<=k<=n, of coefficients of polynomials P_n(x) related to convolution of the k-fold factorials.

%C Let R(m,n,k), 0<=k<=n, the Riordan array (1,x*g(x)) where g(x) is g.f. of the m-fold factorials . Then R(m,n,k) = R(m,n-1,k-1) + Sum_{j, 0<=j<=n-1-k} R(m,n-1,k+j)*P_m(j), R(m,n,0) = 0^n and R(m,0,k) = 0 if k>n.

%F P_0(x) = 1, P_1(x) = x, P_2(x) = 2*x^2, P_ n(x) = n*x*P_(n-1)(x) + Sum_{j, 1<=j<=n-1} j*P_j(x)*P_(n-1-j)(x).

%F P_n(x) = Sum_{k, 0<=k<=n} T(n, k)*x^k.

%F P_n(0) = A000007(n).

%F P_n(x) = A075834(n+1), A111088(n+1), A113130(n+1), A113131(n+1), A113132(n+1), A113133(n+1), A113134(n+1), A113135(n+1) for x = 1, 2, 3, 4, 5, 6, 7, 8 respectively.

%F P_n(-1) = (-1)^n*A000108(n), signed Catalan numbers.

%F T(n, n) = n! = A000142(n).

%F T(2*n+1, n+1) = A000699(n+1) (number of irreducible diagrams with 2n+2 nodes).

%F T(2*n+2, n+2) = A113332(n) = A000699(n+2)*(2*n+3)*(n+2)/(3*(n+1)).

%e Triangle begins:

%e .1;

%e .0, 1;

%e .0, 0, 2;

%e .0, 0, 1, 6;

%e .0, 0, 0, 10, 24;

%e .0, 0, 0, 4, 82, 120;

%e .0, 0, 0, 0, 84, 672, 720;

%e .0, 0, 0, 0, 27, 1236, 5820, 5040;

%e .0, 0, 0, 0, 0, 930, 16328, 54288, 40320;

%e .0, 0, 0, 0, 0, 248, 20850, 211080, 548496, 362880;

%e .0, 0, 0, 0, 0, 0, 12452, 396528, 2775432, 6003360, 362880;

%e .0, 0, 0, 0, 0, 0, 2830, 38732, 7057308, 37831752, 71019360, 39916800;

%Y Cf. A000007, A000108, A000142, A000699.

%Y R(m, n, k) : A097805 (m=0), A084938 (m=1), A111106 (m=2), A113333 (column sums).

%K easy,nonn,tabl

%O 0,6

%A _Philippe Deléham_ and _Paul D. Hanna_, Oct 28 2005

%E Corrected by _Philippe Deléham_, Dec 18 2008