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A213953 Triangle by rows, inverse of A208891.

%I A213953 #13 Jul 21 2015 05:33:54
%S A213953 1,-1,1,0,-1,1,1,-1,-1,1,1,1,-2,-1,1,-2,5,0,-3,-1,1,-9,5,10,-2,-4,-1,
%T A213953 1,-9,-21,25,15,-5,-5,-1,1,50,-105,-11,62,19,-9,-6,-1,1,267,-141,-301,
%U A213953 56,119,21,-14,-7,-1,1,413,777
%N A213953 Triangle by rows, inverse of A208891.
%F A213953 Inverse of triangle A208891, Pascal's triangle matrix with an appended right border of 1's.
%e A213953 Triangle starts:
%e A213953 1;
%e A213953 -1, 1
%e A213953 0, -1, 1
%e A213953 1, -1, -1, 1;
%e A213953 1, 1, -2, -1, 1;
%e A213953 -2, 5, 0, -3, -1, 1;
%e A213953 -9, 5, 10, -2, -4, -1, 1;
%e A213953 -9, -21, 25, 15, -5, -5, -1, 1;
%e A213953 50, -105, -11, 62, 19, -9, -6, -1, 1;
%e A213953 267, -141, -301, 56, 119, 21, -14, -7, -1, 1;
%e A213953 413, 777, -1040, -566, 226, 198, 20, -20, -8, -1, 1;
%e A213953 ...
%p A213953 A208891 := proc(n,k)
%p A213953     if n <0 or k<0 or k>n then
%p A213953             0;
%p A213953     elif n = k then
%p A213953             1 ;
%p A213953     else
%p A213953             binomial(n-1,k) ;
%p A213953     end if;
%p A213953 end proc:
%p A213953 A259456 := proc(n)
%p A213953     local A, row, col ;
%p A213953     A := Matrix(n, n) ;
%p A213953     for row from 1 to n do
%p A213953         for col from 1 to n do
%p A213953             A[row, col] := A208891(row-1,col-1) ;
%p A213953         end do:
%p A213953     end do:
%p A213953     LinearAlgebra[MatrixInverse](A) ;
%p A213953 end proc:
%p A213953 A259456(20) ; # _R. J. Mathar_, Jul 21 2015
%Y A213953 Cf. A208891, A000587 (first column), A014619 (2nd column), A080956 (4th subdiagonal).
%K A213953 tabl,sign
%O A213953 0,13
%A A213953 _Gary W. Adamson_, Jun 26 2012