%I #22 Apr 28 2016 11:29:23

%S 1,2,3,1,0,20,96,155,90,5,-6,0,280,2772,10518,18711,14385,1323,-2863,

%T -126,360,0,2800,47040,323336,1157760,2238855,2050020,207158,-810600,

%U -58505,322740,7956,-45360,0,12320,314160,3409472,20401128,72418826,150057435,154651321,12413874,-101524412,-6408765,82588957,3394248,-37374084,-546480,5443200,0

%N Triangle of numerators of coefficients of the polynomial Q^(2)_m(n) defined by the recursion Q^(2)_0(n)=1; for m>=1, Q^(2)_m(n) = Sum_{i=1..n} i^2*Q^(2)_(m-1)(i). For m>=0, the denominator for all 3*m+1 terms of the m-th row is A202367(m+1).

%C Consider sequence of sequences of polynomials {Q^(0)_m(x)}, {Q^(1)_m(x)},...,{Q^(r)_m(x)},..., such that in every sequence m=0,1,...

%C Sequence {Q^(r)_m(x)} is defined by the recursion: Q^(r)_0(x)=1; for m>=1 and integer x=n, Q^(r)_m(n)=sum{i=1,...,n}i^rQ^(r)_(m-1)(i). By the induction, we see that polynomial Q^(r)_m(x) has degree (r+1)*m. Note that Q^(0)_m(n) is C(n+m-1,m), Q^(1)_m(n)=S(n+m,n), where S(k,l) are Stirling numbers of the second kind. Thus Q^(r)_m(x) is an r-generalization of binomial coefficients and Stirling numbers of the second kind. Moreover, for every r, LCM of denominators of the coefficients of Q^(r)_m(x) generate sequences of factorial type which possess important arithmetic properties. For r=0, it is n!, for r=1, it is A053657, for r=2,3,4 we obtain A202367, A202368, A202369. Denote the general term of the sequence corresponding to a given r by n!^(r) and, for 0<=m<=n, denote C^(r)(n,m)=n!^(r)/(m!^(r)*(n-m)!^(r). Then, for the "r-Pascal triangle", we have the following conjectural regularity: if a prime p==1 mod r, then the ((p-1)/r)-th row contains two 1's and numbers multiple of p. Cf. triangles A202917, A202941.

%F Q^(2)_n(1)=1.

%e The sequence of polynomials begins:

%e Q^(2)_0=1,

%e Q^(2)_1=(2*x^3+3*x^2+x)/6,

%e Q^(2)_2=(20*x^6+96*x^5+155*x^4+90*x^3+5*x^2-6*x)/360,

%e Q^(2)_3=(280*x^9+2772*x^8+10518*x^7+18711*x^6+14385*x^5+1323*x^4-2863*x^3 -126*x^2+360*x)/45360.

%Y Cf. A202339, A053657, A202367, A202368, A202369.

%K sign,tabf

%O 0,2

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Dec 20 2011