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A096044 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^10-M)/9, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

%I A096044 #23 Dec 02 2022 17:25:14
%S A096044 1,11,2,111,33,3,1111,444,66,4,11111,5555,1110,110,5,111111,66666,
%T A096044 16665,2220,165,6,1111111,777777,233331,38885,3885,231,7,11111111,
%U A096044 8888888,3111108,622216,77770,6216,308,8,111111111,99999999,39999996,9333324,1399986,139986,9324,396,9
%N A096044 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^10-M)/9, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.
%H A096044 Alois P. Heinz, <a href="/A096044/b096044.txt">Rows n = 1..141, flattened</a>
%e A096044 Triangle T(n,k) begins:
%e A096044        1;
%e A096044       11,     2;
%e A096044      111,    33,     3;
%e A096044     1111,   444,    66,    4;
%e A096044    11111,  5555,  1110,  110,   5;
%e A096044   111111, 66666, 16665, 2220, 165, 6;
%e A096044   ...
%p A096044 P:= proc(n) option remember; local M; M:= Matrix(n, (i, j)-> binomial(i-1, j-1)); (M^10-M)/9 end: T:= (n, k)-> P(n+1)[n+1, k]: seq(seq(T(n, k), k=1..n), n=1..11);  # _Alois P. Heinz_, Oct 07 2009
%t A096044 P[n_] := P[n] = With[{M = Array[Binomial[#1-1, #2-1]&, {n, n}]}, (MatrixPower[M, 10] - M)/9]; T[n_, k_] := P[n+1][[n+1, k]]; Table[ Table[T[n, k], {k, 1, n}], {n, 1, 11}] // Flatten (* _Jean-François Alcover_, Jan 28 2015, after _Alois P. Heinz_ *)
%Y A096044 Cf. A007318. First column gives A000042. Row sums give A016135.
%Y A096044 Cf. A096034, A096035, A096039, A096040, A096041, A096042, A096043.
%K A096044 nonn,tabl
%O A096044 1,2
%A A096044 _Gary W. Adamson_, Jun 17 2004
%E A096044 Edited and more terms from _Alois P. Heinz_, Oct 07 2009