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A158821 Triangle read by rows: row n (n>=0) ends with 1, and for n>=1 begins with n; other entries are zero.

%I A158821 #17 Apr 05 2023 09:34:45
%S A158821 1,1,1,2,0,1,3,0,0,1,4,0,0,0,1,5,0,0,0,0,1,6,0,0,0,0,0,1,7,0,0,0,0,0,
%T A158821 0,1,8,0,0,0,0,0,0,0,1,9,0,0,0,0,0,0,0,0,1,10,0,0,0,0,0,0,0,0,0,1,11,
%U A158821 0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,1,13,0,0,0,0,0,0,0,0,0,0,0,0,1
%N A158821 Triangle read by rows: row n (n>=0) ends with 1, and for n>=1 begins with n; other entries are zero.
%H A158821 G. C. Greubel, <a href="/A158821/b158821.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A158821 T(n, k) = A145677(n, n-k-1). - _R. J. Mathar_, Apr 01 2009
%F A158821 From _G. C. Greubel_, Dec 22 2021: (Start)
%F A158821 Sum_{k=0..n} T(n, k) = A000027(n).
%F A158821 Sum_{k=0..floor(n/2)} T(n-k, k) = A109613(n). (End)
%e A158821 Triangle begins:
%e A158821   1;
%e A158821   1, 1;
%e A158821   2, 0, 1;
%e A158821   3, 0, 0, 1;
%e A158821   4, 0, 0, 0, 1;
%e A158821   5, 0, 0, 0, 0, 1;
%e A158821   6, 0, 0, 0, 0, 0, 1;
%e A158821   7, 0, 0, 0, 0, 0, 0, 1;
%p A158821 A158821:= proc(n,k)
%p A158821     if n = k then 1;
%p A158821     elif k = 0 then n;
%p A158821     else 0;
%p A158821     end if;
%p A158821 end proc: # _R. J. Mathar_, Jan 08 2015
%t A158821 T[n_, k_]:= If[k==0, Boole[n==0] +n, If[k==n, 1, 0]];
%t A158821 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 22 2021 *)
%t A158821 Join[{1},Table[Join[{n},PadLeft[{1},n,0]],{n,15}]]//Flatten (* _Harvey P. Dale_, Apr 05 2023 *)
%o A158821 (Sage)
%o A158821 def A158821(n,k):
%o A158821     if (k==0): return n + bool(n==0)
%o A158821     elif (k==n): return 1
%o A158821     else: return 0
%o A158821 flatten([[A158821(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Dec 22 2021
%Y A158821 Cf. A000027, A109613, A145677.
%K A158821 nonn,tabl,easy
%O A158821 0,4
%A A158821 _Gary W. Adamson_, Mar 30 2008