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A145677 Triangle T(n, k) read by rows: T(n, 0) = 1, T(n, n) = n, n>0, T(n,k) = 0, 0 < k < n-1.

%I A145677 #14 Jul 24 2023 11:05:25
%S A145677 1,1,1,1,0,2,1,0,0,3,1,0,0,0,4,1,0,0,0,0,5,1,0,0,0,0,0,6,1,0,0,0,0,0,
%T A145677 0,7,1,0,0,0,0,0,0,0,8,1,0,0,0,0,0,0,0,0,9,1,0,0,0,0,0,0,0,0,0,10,1,0,
%U A145677 0,0,0,0,0,0,0,0,0,11
%N A145677 Triangle T(n, k) read by rows: T(n, 0) = 1, T(n, n) = n, n>0, T(n,k) = 0, 0 < k < n-1.
%C A145677 The first entry in each row is 1, the last entry in each of the rows consist of the positive integers (starting 1,1,2,3,...), and all other entries in the triangle are 0's (see example).
%C A145677 The vector of (1, 1, 2, 5, 16, 65, 326,...), which is 1 followed by A000522, is an eigenvector of the matrix: 1 + Sum_{k=1..n} T(n,k)*A000522(k-1) = A000522(n).
%H A145677 G. C. Greubel, <a href="/A145677/b145677.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A145677 T(n, k) = A158821(n,n-k).
%F A145677 1 + Sum_{k= 1..n} T(n,k) *(k-1) = A002061(n).
%F A145677 From _G. C. Greubel_, Dec 23 2021: (Start)
%F A145677 Sum_{k=0..n} T(n, k) = A000027(n).
%F A145677 Sum_{k=0..floor(n/2)} T(n-k, k) = A158416(n) = A152271(n+1). (End)
%e A145677 First few rows of the triangle:
%e A145677   1;
%e A145677   1, 1;
%e A145677   1, 0, 2;
%e A145677   1, 0, 0, 3;
%e A145677   1, 0, 0, 0, 4;
%e A145677   1, 0, 0, 0, 0, 5;
%e A145677   1, 0, 0, 0, 0, 0, 6;
%e A145677   1, 0, 0, 0, 0, 0, 0, 7;
%e A145677   1, 0, 0, 0, 0, 0, 0, 0, 8;
%e A145677   1, 0, 0, 0, 0, 0, 0, 0, 0, 9;
%e A145677   1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10;
%t A145677 T[n_, k_]:= If[k==0, 1, If[k==n, n, 0]];
%t A145677 Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 23 2021 *)
%o A145677 (Sage)
%o A145677 def A145677(n,k):
%o A145677     if (k==0): return 1
%o A145677     elif (k==n): return n
%o A145677     else: return 0
%o A145677 flatten([[A145677(n,k) for k in (0..n)] for n in (0..14)]) # _G. C. Greubel_, Dec 23 2021
%Y A145677 Cf. A128229, A002061, A000522, A152271, A158416.
%K A145677 nonn,tabl,easy
%O A145677 0,6
%A A145677 _Gary W. Adamson_ and _Roger L. Bagula_, Mar 28 2009
%E A145677 Edited by _R. J. Mathar_, Oct 02 2009