A213888 Triangle of coefficients of representations of columns of A213744 in binomial basis.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 0, 0, 5, 10, 10, 5, 1, 0, 0, 4, 15, 20, 15, 6, 1, 0, 0, 3, 18, 35, 35, 21, 7, 1, 0, 0, 2, 19, 52, 70, 56, 28, 8, 1, 0, 0, 1, 18, 68, 121, 126, 84, 36, 9, 1, 0

Offset: 0

Author

Vladimir Shevelev and Peter J. C. Moses, Jun 23 2012

Keywords

• nonn
• tabl

Comments

This triangle is the fourth array in the sequence of arrays A026729, A071675, A213887,..., such that the first two arrays are 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 A213744. For example, s_1(n)=binomial(n,1)=n is the first column of A213744 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213744 for n>1, etc. In particular (see comment inA213744), in cases k=7,8,9 s_k(n) is A063262(n+2), A063263(n+2), A063264(n+2) respectively.

Examples

			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....1....4....6....4....1
.6..|..0....0....5...10...10....5....1
.7..|..0....0....4...15...20...15....6....1
.8..|..0....0....3...18...35...35...21....7....1
.9..|..0....0....2...19...52...70...56...28....8....1
		

Crossrefs

Cf. A026729, A071675, A213887, A030528 (parts <=2), A078803 (parts <=3), A213887 (parts <=4).

Programs

Maple

pts := 5; # A213888
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