A175669 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).

1, 2, 3, 1, 0, 20, 96, 155, 90, 5, -6, 0, 280, 2772, 10518, 18711, 14385, 1323, -2863, -126, 360, 0, 2800, 47040, 323336, 1157760, 2238855, 2050020, 207158, -810600, -58505, 322740, 7956, -45360, 0, 12320, 314160, 3409472, 20401128, 72418826, 150057435, 154651321, 12413874, -101524412, -6408765, 82588957, 3394248, -37374084, -546480, 5443200, 0

Offset: 0

Author

Vladimir Shevelev and Peter J. C. Moses, Dec 20 2011

Keywords

• sign
• tabf

Comments

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,...

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.

Examples

			The sequence of polynomials begins:
Q^(2)_0=1,
Q^(2)_1=(2*x^3+3*x^2+x)/6,
Q^(2)_2=(20*x^6+96*x^5+155*x^4+90*x^3+5*x^2-6*x)/360,
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.
		

Crossrefs

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

Formula

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