A213887 - OEIS

A213887

Triangle of coefficients of representations of columns of A213743 in binomial basis.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 0, 4, 6, 4, 1, 0, 0, 3, 10, 10, 5, 1, 0, 0, 2, 12, 20, 15, 6, 1, 0, 0, 1, 12, 31, 35, 21, 7, 1, 0, 0, 0, 10, 40, 65, 56, 28, 8, 1, 0, 0, 0, 6, 44, 101, 120, 84, 36, 9, 1, 0

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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

LINKS

Table of n, a(n) for n=0..66.

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