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A132615 Triangle T, read by rows, where row n+1 of T = row n of T^(2n-1) with appended '1' for n>=0 with T(0,0)=1.

%I A132615 #16 Apr 27 2020 06:21:00
%S A132615 1,1,1,1,1,1,6,3,1,1,80,25,5,1,1,1666,378,56,7,1,1,47232,8460,1020,99,
%T A132615 9,1,1,1694704,252087,26015,2134,154,11,1,1,73552752,9392890,855478,
%U A132615 61919,3848,221,13,1,1,3744491970,420142350,34461260,2257413,125760,6290,300,15,1,1
%N A132615 Triangle T, read by rows, where row n+1 of T = row n of T^(2n-1) with appended '1' for n>=0 with T(0,0)=1.
%H A132615 Alois P. Heinz, <a href="/A132615/b132615.txt">Rows n = 0..140, flattened</a>
%F A132615 T(n+1,1) is divisible by 2n-1 for n>=1.
%e A132615 Triangle begins:
%e A132615   1;
%e A132615   1, 1;
%e A132615   1, 1, 1;
%e A132615   6, 3, 1, 1;
%e A132615   80, 25, 5, 1, 1;
%e A132615   1666, 378, 56, 7, 1, 1;
%e A132615   47232, 8460, 1020, 99, 9, 1, 1;
%e A132615   1694704, 252087, 26015, 2134, 154, 11, 1, 1;
%e A132615   73552752, 9392890, 855478, 61919, 3848, 221, 13, 1, 1; ...
%e A132615 GENERATE T FROM ODD MATRIX POWERS OF T.
%e A132615 Matrix cube, T^3, begins:
%e A132615   1;
%e A132615   3, 1;
%e A132615   6, 3, 1; <-- row 3 of T
%e A132615   31, 12, 3, 1;
%e A132615   357, 100, 18, 3, 1;
%e A132615   6786, 1455, 205, 24, 3, 1; ...
%e A132615 where row 3 of T = row 2 of T^3 with appended '1'.
%e A132615 Matrix fifth power, T^5, begins:
%e A132615   1;
%e A132615   5, 1;
%e A132615   15, 5, 1;
%e A132615   80, 25, 5, 1; <-- row 4 of T
%e A132615   855, 215, 35, 5, 1;
%e A132615   15171, 3065, 410, 45, 5, 1; ...
%e A132615 where row 4 of T = row 3 of T^5 with appended '1'.
%e A132615 Matrix seventh power, T^7, begins:
%e A132615   1;
%e A132615   7, 1;
%e A132615   28, 7, 1;
%e A132615   161, 42, 7, 1;
%e A132615   1666, 378, 56, 7, 1; <-- row 5 of T
%e A132615   28119, 5348, 679, 70, 7, 1; ...
%e A132615 where row 5 of T = row 4 of T^7 with appended '1'.
%e A132615 ALTERNATE GENERATING METHOD.
%e A132615 Row 4: start with a '1' followed by 4 zeros;
%e A132615 take partial sums and append 2 zeros; then
%e A132615 take partial sums thrice more:
%e A132615   (1), 0, 0, 0, 0;
%e A132615   1, 1, 1, 1, (1), 0, 0;
%e A132615   1, 2, 3, 4, 5, 5, (5);
%e A132615   1, 3, 6, 10, 15, 20, (25);
%e A132615   1, 4, 10, 20, 35, 55, (80);
%e A132615 the final nonzero terms form row 4: [80, 25, 5, 1, 1].
%e A132615 Row 5: start with a '1' followed by 6 zeros;
%e A132615 take partial sums and append 4 zeros;
%e A132615 take partial sums and append 2 zeros; then
%e A132615 take partial sums thrice more:
%e A132615   (1), 0, 0, 0, 0, 0, 0;
%e A132615   1, 1, 1, 1, 1, 1, (1), 0, 0, 0, 0;
%e A132615   1, 2, 3, 4, 5, 6, 7, 7, 7, 7, (7), 0, 0;
%e A132615   1, 3, 6, 10, 15, 21, 28, 35, 42, 49, 56, 56, (56);
%e A132615   1, 4, 10, 20, 35, 56, 84, 119, 161, 210, 266, 322, (378);
%e A132615   1, 5, 15, 35, 70, 126, 210, 329, 490, 700, 966, 1288, (1666);
%e A132615 the final nonzero terms form row 5: [1666, 378, 56, 7, 1, 1].
%e A132615 Continuing in this way produces all the rows of this triangle.
%p A132615 b:= proc(n) option remember;
%p A132615       Matrix(n, (i,j)-> T(i-1,j-1))^(2*n-3)
%p A132615     end:
%p A132615 T:= proc(n,k) option remember;
%p A132615      `if`(n=k, 1, `if`(k>n, 0, b(n)[n,k+1]))
%p A132615     end:
%p A132615 seq(seq(T(n,k), k=0..n), n=0..10);  # _Alois P. Heinz_, Apr 13 2020
%t A132615 b[n_] := b[n] = MatrixPower[Table[T[i-1, j-1], {i, n}, {j, n}], 2n-3];
%t A132615 T[n_, k_] := T[n, k] = If[n == k, 1, If[k > n, 0, b[n][[n, k + 1]]]];
%t A132615 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 27 2020, after _Alois P. Heinz_ *)
%o A132615 (PARI) {T(n, k)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-k-1, p=(n-1)*(n-2)-(n-j-1)*(n-j-2); A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A=Vec((Polrev(A)+x*O(x^p))/(1-x)); A[p+1]}
%Y A132615 Cf. columns: A132616, A132617, A132618; A132619; variants: A132610, A101479.
%Y A132615 Cf. A304190, A304191, A304193.
%K A132615 nonn,tabl
%O A132615 0,7
%A A132615 _Paul D. Hanna_, Aug 24 2007