OFFSET
0,9
COMMENTS
This triangle is the third array in the sequence of arrays A026729, A071675 considered as triangles.
Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th row of the triangle. Then s_k(n)=sum{i=0,...,k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A213743. For example, s_1(n)=binomial(n,1)=n is the first column of A213743 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213743 for n>1, etc. In particular (see comment in A213743), in cases k=6,7,8,9 s_k(n) is A064056(n+2), A064057(n+2), A064058(n+2), A000575(n+3) respectively.
Riordan array (1,x+x^2+x^3+x^4). A186332 with additional 0 column. - Ralf Stephan, Dec 31 2013
EXAMPLE
As a triangle, this begins
n/k.|..0....1....2....3....4....5....6....7....8....9
=====================================================
.0..|..1
.1..|..0....1
.2..|..0....1....1
.3..|..0....1....2....1
.4..|..0....1....3....3....1
.5..|..0....0....4....6....4....1
.6..|..0....0....3...10...10....5....1
.7..|..0....0....2...12...20...15....6....1
.8..|..0....0....1...12...31...35...21....7....1
.9..|..0....0....0...10...40...65...56...28....8....1
MAPLE
pts := 4; # A213887
g := 1/(1-t*z*add(z^i, i=0..pts-1)) ;
for n from 0 to 13 do
for k from 0 to n do
coeftayl(g, z=0, n) ;
coeftayl(%, t=0, k) ;
printf("%d ", %) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, May 28 2025
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Shevelev and Peter J. C. Moses, Jun 23 2012
STATUS
approved
