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A128132 A natural number transform, companion to A127701.

%I A128132 #22 Jul 27 2020 04:58:55
%S A128132 1,-1,2,0,-1,3,0,0,-1,4,0,0,0,-1,5,0,0,0,0,-1,6,0,0,0,0,0,-1,7,0,0,0,
%T A128132 0,0,0,-1,8,0,0,0,0,0,0,0,-1,9,0,0,0,0,0,0,0,0,-1,10,0,0,0,0,0,0,0,0,
%U A128132 0,-1,11,0,0,0,0,0,0,0,0,0,0,-1,12
%N A128132 A natural number transform, companion to A127701.
%C A128132 Binomial transform is A128133.
%F A128132 T(n,n) = n.
%F A128132 T(n,n-1) = -1.
%F A128132 T(n,k) = 0 for k <> n, n-1.
%e A128132 Triangle T(n,k) (with rows n >= 1 and columns k = 1..n) begins:
%e A128132    1;
%e A128132   -1,  2;
%e A128132    0, -1,  3;
%e A128132    0,  0, -1,  4;
%e A128132    0,  0,  0, -1,  5;
%e A128132    0,  0,  0,  0, -1, 6;
%e A128132    ...
%p A128132 A128132 := proc(n,k)
%p A128132     if n = k then
%p A128132         n;
%p A128132     elif k = n-1 then
%p A128132         -1 ;
%p A128132     else
%p A128132         0 ;
%p A128132     end if;
%p A128132 end proc: # _R. J. Mathar_, Apr 26 2016
%t A128132 {1}~Join~Table[PadLeft[{-1, n}, n], {n, 2, 12}] // Flatten (* _Michael De Vlieger_, Apr 26 2016 *)
%Y A128132 Cf. A127701, A128133, A128134.
%K A128132 tabl,sign,easy
%O A128132 1,3
%A A128132 _Gary W. Adamson_, Feb 15 2007