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%I #27 Feb 16 2022 11:04:58
%S 1,2,2,3,8,3,4,22,22,4,5,52,82,52,5,6,114,254,254,114,6,7,240,677,
%T 1000,677,240,7,8,494,1692,3176,3176,1692,494,8,9,1004,3972,9136,
%U 12182,9136,3972,1004,9,10,2026,9052,24202,40564,40564,24202,9052,2026,10,11,4072,19975,60828,123414,155096,123414,60828,19975,4072,11
%N Lower triangular matrix T with first column and diagonal (1,2,3,4,...,n,...) and otherwise satisfying T(i,j) = Sum_{k=1..j} T(i-j+1,k)*T(j,k), read by rows.
%C The definition is equivalent to requiring that L'=L*Transpose(L), where L' is formed from L by shifting column j upward j-1 rows for all j. If the first column is (1,1,1,1,...,1,...} then the lower triangular matrix contains Pascal's triangle. Column two and one-half of column two are essentially A005803 (second-order Eulerian numbers 2^n - 2*n) and A000295 (Eulerian numbers 2^n - n - 1), respectively. Column three is A135836.
%H Michel Marcus, <a href="/A135835/b135835.txt">Rows n = 1..50 of triangle, flattened</a>
%H Alan Edelman and Gilbert Strang, <a href="http://www.jstor.org/stable/4145127">Pascal Matrices</a>, Am. Math. Monthly 111 (2004) 189-197.
%F T(n, 1) = T(n, n) = n, otherwise T(n,k) = Sum_{j=1..k} T(n-k+1, j)*T(k, j).
%F T(n, n-k) = T(n, k).
%F T(n, 2) = A005803(n+1) = 2*A000295(n).
%F T(n, 3) = A135836(n-2).
%e From _Philippe Deléham_, Oct 10 2011: (Start)
%e Triangle begins:
%e 1;
%e 2, 2;
%e 3, 8, 3;
%e 4, 22, 22, 4;
%e 5, 52, 82, 52, 5;
%e 6, 114, 254, 254, 114, 6;
%e ...
%e (End)
%t T[n_, k_]:= T[n, k]= If[k<1 || k>n, 0, If[k==1 || k==n, n, Sum[T[n-k+1, j]*T[k, j], {j, k}]]];
%t Table[T[n, k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Feb 07 2022 *)
%o (PARI) T(i,j) = if (j>i, 0, if ((j==1) || (i==j), i, sum(k=1, j, T(i-j+1,k)*T(j,k))));
%o tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print()); \\ _Michel Marcus_, Sep 30 2017
%o (Sage)
%o @CachedFunction
%o def T(n,k): # A135835
%o if (k<0 or k>n): return 0
%o elif (k==1 or k==n): return n
%o else: return sum( T(n-k+1,j)*T(k,j) for j in (1..k) )
%o flatten([[T(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Feb 07 2022
%Y Cf. A000295, A005803, A135836.
%K nonn,tabl
%O 1,2
%A _John W. Layman_, Nov 30 2007
%E Name and formula corrected, and more terms from _Michel Marcus_, Sep 30 2017
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