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%I #14 Jul 24 2023 11:05:25
%S 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 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 0,0,0,0,0,0,0,0,0,11
%N 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 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 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 G. C. Greubel, <a href="/A145677/b145677.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = A158821(n,n-k).
%F 1 + Sum_{k= 1..n} T(n,k) *(k-1) = A002061(n).
%F From _G. C. Greubel_, Dec 23 2021: (Start)
%F Sum_{k=0..n} T(n, k) = A000027(n).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A158416(n) = A152271(n+1). (End)
%e First few rows of the triangle:
%e 1;
%e 1, 1;
%e 1, 0, 2;
%e 1, 0, 0, 3;
%e 1, 0, 0, 0, 4;
%e 1, 0, 0, 0, 0, 5;
%e 1, 0, 0, 0, 0, 0, 6;
%e 1, 0, 0, 0, 0, 0, 0, 7;
%e 1, 0, 0, 0, 0, 0, 0, 0, 8;
%e 1, 0, 0, 0, 0, 0, 0, 0, 0, 9;
%e 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10;
%t T[n_, k_]:= If[k==0, 1, If[k==n, n, 0]];
%t Table[T[n, k], {n,0,14}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 23 2021 *)
%o (Sage)
%o def A145677(n,k):
%o if (k==0): return 1
%o elif (k==n): return n
%o else: return 0
%o flatten([[A145677(n,k) for k in (0..n)] for n in (0..14)]) # _G. C. Greubel_, Dec 23 2021
%Y Cf. A128229, A002061, A000522, A152271, A158416.
%K nonn,tabl,easy
%O 0,6
%A _Gary W. Adamson_ and _Roger L. Bagula_, Mar 28 2009
%E Edited by _R. J. Mathar_, Oct 02 2009
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