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A132610 Triangle T, read by rows, where row n+1 of T = row n of matrix power T^(2n) with appended '1' for n>=0 with T(0,0)=1.

%I A132610 #19 Apr 27 2020 05:55:55
%S A132610 1,1,1,2,1,1,14,4,1,1,194,39,6,1,1,4114,648,76,8,1,1,118042,15465,
%T A132610 1510,125,10,1,1,4274612,483240,41121,2908,186,12,1,1,186932958,
%U A132610 18685905,1424178,89670,4970,259,14,1,1,9577713250,861282832,59857416,3437248,171700,7824,344,16,1,1
%N A132610 Triangle T, read by rows, where row n+1 of T = row n of matrix power T^(2n) with appended '1' for n>=0 with T(0,0)=1.
%H A132610 Alois P. Heinz, <a href="/A132610/b132610.txt">Rows n = 0..140, flattened</a>
%F A132610 T(n+1,1) is divisible by n for n>=1.
%e A132610 Triangle begins:
%e A132610 1;
%e A132610 1, 1;
%e A132610 2, 1, 1;
%e A132610 14, 4, 1, 1;
%e A132610 194, 39, 6, 1, 1;
%e A132610 4114, 648, 76, 8, 1, 1;
%e A132610 118042, 15465, 1510, 125, 10, 1, 1;
%e A132610 4274612, 483240, 41121, 2908, 186, 12, 1, 1;
%e A132610 186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1; ...
%e A132610 GENERATE T FROM EVEN MATRIX POWERS OF T.
%e A132610 Matrix square T^2 begins:
%e A132610 1;
%e A132610 2, 1; <-- row 2 of T
%e A132610 5, 2, 1;
%e A132610 34, 9, 2, 1;
%e A132610 453, 88, 13, 2, 1; ...
%e A132610 where row 2 of T = row 1 of T^2 with appended '1'.
%e A132610 Matrix fourth powers T^4 begins:
%e A132610 1;
%e A132610 4, 1;
%e A132610 14, 4, 1; <-- row 3 of T
%e A132610 96, 22, 4, 1;
%e A132610 1215, 220, 30, 4, 1; ...
%e A132610 where row 3 of T = row 2 of T^4 with appended '1'.
%e A132610 Matrix sixth power T^6 begins:
%e A132610 1;
%e A132610 6, 1;
%e A132610 27, 6, 1;
%e A132610 194, 39, 6, 1; <-- row 4 of T
%e A132610 2394, 404, 51, 6, 1; ...
%e A132610 where row 4 of T = row 3 of T^6 with appended '1'.
%e A132610 ALTERNATE GENERATING METHOD.
%e A132610 Start with [1,0,0,0]; take partial sums and append 1 zero;
%e A132610 take partial sums twice more:
%e A132610 (1), 0, 0, 0;
%e A132610 1, 1, 1, (1), 0;
%e A132610 1, 2, 3, 4, (4);
%e A132610 1, 3, 6, 10, (14);
%e A132610 the final nonzero terms form row 3: [14,4,1,1].
%e A132610 Start with [1,0,0,0,0,0]; take partial sums and append 3 zeros;
%e A132610 take partial sums and append 1 zero; take partial sums twice more:
%e A132610 (1), 0, 0, 0, 0, 0;
%e A132610 1, 1, 1, 1, 1, (1), 0, 0, 0;
%e A132610 1, 2, 3, 4, 5, 6, 6, 6, (6), 0;
%e A132610 1, 3, 6, 10, 15, 21, 27, 33, 39, (39);
%e A132610 1, 4, 10, 20, 35, 56, 83, 116, 155, (194);
%e A132610 the final nonzero terms form row 4: [194,39,6,1,1].
%e A132610 Continuing in this way produces all the rows of this triangle.
%p A132610 b:= proc(n) option remember;
%p A132610       Matrix(n, (i,j)-> T(i-1,j-1))^(2*n-2)
%p A132610     end:
%p A132610 T:= proc(n,k) option remember;
%p A132610      `if`(n=k, 1, `if`(k>n, 0, b(n)[n,k+1]))
%p A132610     end:
%p A132610 seq(seq(T(n,k), k=0..n), n=0..10);  # _Alois P. Heinz_, Apr 13 2020
%t A132610 b[n_] := b[n] = MatrixPower[Table[T[i-1, j-1], {i, n}, {j, n}], 2n-2];
%t A132610 T[n_, k_] := T[n, k] = If[n == k, 1, If[k > n, 0, b[n][[n, k+1]]]];
%t A132610 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 27 2020, after _Alois P. Heinz_ *)
%o A132610 (PARI) {T(n, k)=local(A=vector(n+1), p); A[1]=1; for(j=1, n-k-1, p=(n-1)^2-(n-j-1)^2; A=Vec((Polrev(A)+x*O(x^p))/(1-x))); A=Vec((Polrev(A)+x*O(x^p))/(1-x)); A[p+1]}
%o A132610 for(n=0,10, for(k=0,n, print1(T(n,k),", "));print(""))
%Y A132610 Cf. columns: A132611, A132612, A132613; A132614; variants: A132615, A101479.
%Y A132610 Cf. A304189, A304192, A304188.
%K A132610 nonn,tabl
%O A132610 0,4
%A A132610 _Paul D. Hanna_, Aug 23 2007