A213887 - OEIS

%I #19 Dec 31 2013 05:30:47

%S 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,

%T 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,

%U 101,120,84,36,9,1,0

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

%C This triangle is the third array in the sequence of arrays A026729, A071675 considered as triangles.

%C 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.

%C Riordan array (1,x+x^2+x^3+x^4). A186332 with additional 0 column. - _Ralf Stephan_, Dec 31 2013

%e As a triangle, this begins

%e n/k.|..0....1....2....3....4....5....6....7....8....9

%e =====================================================

%e .0..|..1

%e .1..|..0....1

%e .2..|..0....1....1

%e .3..|..0....1....2....1

%e .4..|..0....1....3....3....1

%e .5..|..0....0....4....6....4....1

%e .6..|..0....0....3...10...10....5....1

%e .7..|..0....0....2...12...20...15....6....1

%e .8..|..0....0....1...12...31...35...21....7....1

%e .9..|..0....0....0...10...40...65...56...28....8....1

%Y Cf. A026729, A071675.

%K nonn,tabl

%O 0,9

%A _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 23 2012