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A213889 Triangle of coefficients of representations of columns of A213745 in binomial basis.

%I A213889 #20 May 28 2025 04:49:47
%S A213889 1,0,1,0,1,1,0,1,2,1,0,1,3,3,1,0,1,4,6,4,1,0,1,5,10,10,5,1,0,0,6,15,
%T A213889 20,15,6,1,0,0,5,21,35,35,21,7,1,0,0,4,25,56,70,56,28,8,1,0,0,3,27,80,
%U A213889 126,126,84,36,9,1
%N A213889 Triangle of coefficients of representations of columns of A213745 in binomial basis.
%C A213889 This array is the fifth array in the sequence of arrays A026729, A071675, A213887, A213888,..., such that the first two arrays are considered as triangles.
%C A213889 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 A213745. For example, s_1(n)=binomial(n,1)=n is the first column of A213745 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213745 for n>1, etc. In particular (see comment in A213745), in cases k=8,9  s_k(n) is A063417(n+2), A063418(n+2) respectively.
%e A213889 As a triangle, this begins
%e A213889 n/k.|..0....1....2....3....4....5....6....7....8....9
%e A213889 =====================================================
%e A213889 .0..|..1
%e A213889 .1..|..0....1
%e A213889 .2..|..0....1....1
%e A213889 .3..|..0....1....2....1
%e A213889 .4..|..0....1....3....3....1
%e A213889 .5..|..0....1....4....6....4....1
%e A213889 .6..|..0....1....5...10...10....5....1
%e A213889 .7..|..0....0....6...15...20...15....6....1
%e A213889 .8..|..0....0....5...21...35...35...21....7....1
%e A213889 .9..|..0....0....4...25...56...70...56...28....8....1
%p A213889 pts := 6; # A213889 and A061676
%p A213889 g := 1/(1-t*z*add(z^i,i=0..pts-1)) ;
%p A213889 for n from 0 to 13 do
%p A213889     for k from 0 to n do
%p A213889         coeftayl(g,z=0,n) ;
%p A213889         coeftayl(%,t=0,k) ;
%p A213889         printf("%d ",%) ;
%p A213889     end do:
%p A213889     printf("\n") ;
%p A213889 end do: # _R. J. Mathar_, May 28 2025
%Y A213889 Cf. A026729, A071675, A078803 (parts <=3), A213887 (parts <=4), A213888 (parts <=5).
%Y A213889 Essentially the same as A061676.
%K A213889 nonn,tabl
%O A213889 0,9
%A A213889 _Vladimir Shevelev_ and _Peter J. C. Moses_, Jun 23 2012