

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)_(m1)(i). For m>=0, the denominator for all 3*m+1 terms of the mth row is A202367(m+1).


9



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
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OFFSET

0,2


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)_(m1)(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+m1,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 rgeneralization 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)*(nm)!^(r). Then, for the "rPascal triangle", we have the following conjectural regularity: if a prime p==1 mod r, then the ((p1)/r)th row contains two 1's and numbers multiple of p. Cf. triangles A202917, A202941.


LINKS

Table of n, a(n) for n=0..50.


FORMULA

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


EXAMPLE

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^26*x)/360,
Q^(2)_3=(280*x^9+2772*x^8+10518*x^7+18711*x^6+14385*x^5+1323*x^42863*x^3 126*x^2+360*x)/45360.


CROSSREFS

Cf. A202339, A053657, A202367, A202368, A202369.
Sequence in context: A137329 A265604 A171996 * A288839 A286583 A321931
Adjacent sequences: A175666 A175667 A175668 * A175670 A175671 A175672


KEYWORD

sign,tabf


AUTHOR

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


STATUS

approved



