OFFSET
0,2
COMMENTS
In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is ((1-z)/(1-2*z)^2)/(1-x*z/(1-z)).
This is the second member of the family of Riordan-type matrices obtained from A007318(n,m) (Pascal's triangle read as lower triangular matrix) by repeated application of the prs-procedure.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1274
FORMULA
a(n, m) = Sum_{k=m,..,n} ( A055248(n, k) ), n >= m >= 0, a(n, m) := 0 if n<m, (sequence of partial row sums in column m).
Column m recursion: a(n, m) = Sum_{j=m,..,(n-1)} ( a(j, m) ) + A055248(n, m), n >= m >= 0, a(n, m) := 0 if n<m.
G.f. for column m: ((1-x)/(1-2*x)^2)*(x/(1-x))^m, m >= 0.
a(n, m) = binomial(n, m) * 2F1(2, m-n; m+1; -1) where 2F1 is the hypergeometric function. Jean-François Alcover, Mar 11 2014
EXAMPLE
1;
3,1;
8,4,1;
20,12,5,1;
...
Fourth row polynomial (n=3): p(3,x)= 20+12*x+5*x^2+x^3
MATHEMATICA
a[n_, m_] := Binomial[n, m]*Hypergeometric2F1[2, m-n, m+1, -1]; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Mar 11 2014 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, May 26 2000
STATUS
approved