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A128229 A natural number transform, inverse of signed A094587.

%I A128229 #21 Mar 13 2021 00:55:55
%S A128229 1,1,1,0,2,1,0,0,3,1,0,0,0,4,1,0,0,0,0,5,1,0,0,0,0,0,6,1,0,0,0,0,0,0,
%T A128229 7,1,0,0,0,0,0,0,0,8,1,0,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,0,0,0,10,1,0,0,
%U A128229 0,0,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,0,0,0,12,1
%N A128229 A natural number transform, inverse of signed A094587.
%C A128229 Signed version of the transform (with -1, -2, -3, ... in the subdiagonal) gives A094587 having row sums A000522: (1, 2, 5, 16, 65, 236, ...). Unsigned inverse gives signed A094587 (with alternate signs); giving row sums = a signed variation of A094587 as follows: (1, 0, 1, -2, 9, -44, 265, -1854, ...). Binomial transform of the triangle = A093375.
%C A128229 Eigensequence of the triangle = A000085 starting (1, 2, 4, 10, 26, 76, ...). - _Gary W. Adamson_, Dec 29 2008
%F A128229 Infinite lower triangular matrix with (1,1,1,...) in the main diagonal and (1,2,3,...) in the subdiagonal.
%F A128229 T(n,n)=1, T(n,n-1)=n-1 and T(n,k)=0 for 1<=k<=n, 1<=n. - _Hartmut F. W. Hoft_, Jun 10 2017
%e A128229 First few rows of the triangle are:
%e A128229 1;
%e A128229 1, 1;
%e A128229 0, 2, 1;
%e A128229 0, 0, 3, 1;
%e A128229 0, 0, 0, 4, 1;
%e A128229 0, 0, 0, 0, 5, 1;
%e A128229 0, 0, 0, 0, 0, 6, 1;
%e A128229 0, 0, 0, 0, 0, 0, 7, 1;
%e A128229 ...
%t A128229 a128229[n_] := Table[Which[r==q, 1, r-1==q, q, True, 0], {r, 1, n}, {q, 1, r}]
%t A128229 Flatten[a128229[13]] (* data *)
%t A128229 TableForm[a128229[8]] (* triangle *)
%t A128229 (* _Hartmut F. W. Hoft_, Jun 10 2017 *)
%o A128229 (Python)
%o A128229 def T(n, k): return 1 if n==k else n - 1 if k==n - 1 else 0
%o A128229 for n in range(1, 11): print([T(n, k) for k in range(1, n + 1)]) # _Indranil Ghosh_, Jun 10 2017
%Y A128229 Cf. A094587, A000522, A094587, A093375, A128228, A128227.
%Y A128229 Cf. A000085. - _Gary W. Adamson_, Dec 29 2008
%K A128229 nonn,tabl
%O A128229 1,5
%A A128229 _Gary W. Adamson_, Feb 19 2007