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.
1, 1, 1, 2, 1, 1, 14, 4, 1, 1, 194, 39, 6, 1, 1, 4114, 648, 76, 8, 1, 1, 118042, 15465, 1510, 125, 10, 1, 1, 4274612, 483240, 41121, 2908, 186, 12, 1, 1, 186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1, 9577713250, 861282832, 59857416, 3437248, 171700, 7824, 344, 16, 1, 1
Offset: 0
Examples
Triangle begins: 1; 1, 1; 2, 1, 1; 14, 4, 1, 1; 194, 39, 6, 1, 1; 4114, 648, 76, 8, 1, 1; 118042, 15465, 1510, 125, 10, 1, 1; 4274612, 483240, 41121, 2908, 186, 12, 1, 1; 186932958, 18685905, 1424178, 89670, 4970, 259, 14, 1, 1; ... GENERATE T FROM EVEN MATRIX POWERS OF T. Matrix square T^2 begins: 1; 2, 1; <-- row 2 of T 5, 2, 1; 34, 9, 2, 1; 453, 88, 13, 2, 1; ... where row 2 of T = row 1 of T^2 with appended '1'. Matrix fourth powers T^4 begins: 1; 4, 1; 14, 4, 1; <-- row 3 of T 96, 22, 4, 1; 1215, 220, 30, 4, 1; ... where row 3 of T = row 2 of T^4 with appended '1'. Matrix sixth power T^6 begins: 1; 6, 1; 27, 6, 1; 194, 39, 6, 1; <-- row 4 of T 2394, 404, 51, 6, 1; ... where row 4 of T = row 3 of T^6 with appended '1'. ALTERNATE GENERATING METHOD. Start with [1,0,0,0]; take partial sums and append 1 zero; take partial sums twice more: (1), 0, 0, 0; 1, 1, 1, (1), 0; 1, 2, 3, 4, (4); 1, 3, 6, 10, (14); the final nonzero terms form row 3: [14,4,1,1]. Start with [1,0,0,0,0,0]; take partial sums and append 3 zeros; take partial sums and append 1 zero; take partial sums twice more: (1), 0, 0, 0, 0, 0; 1, 1, 1, 1, 1, (1), 0, 0, 0; 1, 2, 3, 4, 5, 6, 6, 6, (6), 0; 1, 3, 6, 10, 15, 21, 27, 33, 39, (39); 1, 4, 10, 20, 35, 56, 83, 116, 155, (194); the final nonzero terms form row 4: [194,39,6,1,1]. Continuing in this way produces all the rows of this triangle.
Links
- Alois P. Heinz, Rows n = 0..140, flattened
Crossrefs
Programs
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Maple
b:= proc(n) option remember; Matrix(n, (i,j)-> T(i-1,j-1))^(2*n-2) end: T:= proc(n,k) option remember; `if`(n=k, 1, `if`(k>n, 0, b(n)[n,k+1])) end: seq(seq(T(n,k), k=0..n), n=0..10); # Alois P. Heinz, Apr 13 2020 -
Mathematica
b[n_] := b[n] = MatrixPower[Table[T[i-1, j-1], {i, n}, {j, n}], 2n-2]; T[n_, k_] := T[n, k] = If[n == k, 1, If[k > n, 0, b[n][[n, k+1]]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 27 2020, after Alois P. Heinz *) -
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]} for(n=0,10, for(k=0,n, print1(T(n,k),", "));print(""))
Formula
T(n+1,1) is divisible by n for n>=1.