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A213887 Triangle of coefficients of representations of columns of A213743 in binomial basis.

%I A213887 #22 May 28 2025 04:42:30
%S A213887 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 A213887 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 A213887 101,120,84,36,9,1,0
%N A213887 Triangle of coefficients of representations of columns of A213743 in binomial basis.
%C A213887 This triangle is the third array in the sequence of arrays A026729, A071675 considered as triangles.
%C A213887 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 A213887 Riordan array (1,x+x^2+x^3+x^4). A186332 with additional 0 column. - _Ralf Stephan_, Dec 31 2013
%e A213887 As a triangle, this begins
%e A213887 n/k.|..0....1....2....3....4....5....6....7....8....9
%e A213887 =====================================================
%e A213887 .0..|..1
%e A213887 .1..|..0....1
%e A213887 .2..|..0....1....1
%e A213887 .3..|..0....1....2....1
%e A213887 .4..|..0....1....3....3....1
%e A213887 .5..|..0....0....4....6....4....1
%e A213887 .6..|..0....0....3...10...10....5....1
%e A213887 .7..|..0....0....2...12...20...15....6....1
%e A213887 .8..|..0....0....1...12...31...35...21....7....1
%e A213887 .9..|..0....0....0...10...40...65...56...28....8....1
%p A213887 pts := 4; # A213887
%p A213887 g := 1/(1-t*z*add(z^i,i=0..pts-1)) ;
%p A213887 for n from 0 to 13 do
%p A213887     for k from 0 to n do
%p A213887         coeftayl(g,z=0,n) ;
%p A213887         coeftayl(%,t=0,k) ;
%p A213887         printf("%d ",%) ;
%p A213887     end do:
%p A213887     printf("\n") ;
%p A213887 end do: # _R. J. Mathar_, May 28 2025
%Y A213887 Cf. A026729, A071675, A030528 (parts <=2), A078803 (parts <=3), A213888 (parts <=5), A061676 and A213889 (parts <=6).
%K A213887 nonn,tabl
%O A213887 0,9
%A A213887 _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 23 2012