OFFSET
0,4
COMMENTS
LINKS
G. C. Greubel, Rows n = 0..100 of triangle, flattened
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv:1107.5490 [math.CO], 2011.
Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
FORMULA
Lower triangular (infinite) matrix T = (U - D*P)^{-1} with the unit matrix U, the Pascal matrix P from A007318 and the matrix D with elements delta_{i,j+1}, for i, j >= 0 (row 0 has only 0s). From the Paul Barry paper rewritten in matrix notation. T satisfies P*T = D'*(T - U), with D' the transposed matrix D, that is the diagonal of T has been erased and the row index shifted on the r.h.s. (showing that the name Eigentriangle or -matrix is a misnomer). For finite N X N matrices P*T = D'*(T - U), only up to the last row. - Wolfdieter Lang, Apr 07 2021
EXAMPLE
Triangle T begins
1;
1, 1;
2, 1, 1;
5, 3, 1, 1;
15, 9, 4, 1, 1;
52, 31, 14, 5, 1, 1;
203, 121, 54, 20, 6, 1, 1;
877, 523, 233, 85, 27, 7, 1, 1;
4140, 2469, 1101, 400, 125, 35, 8, 1, 1;
21147, 12611, 5625, 2046, 635, 175, 44, 9, 1, 1;
115975, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1, 1;
Inverse is the identity matrix I minus binomial matrix B shifted down once, or
T^{-1}(n,k)=if(k=n,1,if(k<n,-binomial(n-1,k),0)). This begins
1;
-1, 1;
-1, -1, 1;
-1, -2, -1, 1;
-1, -3, -3, -1, 1;
-1, -4, -6, -4, -1, 1;
-1, -5, -10, -10, -5, -1, 1;
-1, -6, -15, -20, -15, -6, -1, 1;
-1, -7, -21, -35, -35, -21, -7, -1, 1;
-1, -8, -28, -56, -70, -56, -28, -8, -1, 1;
Production matrix is
1, 1;
1, 0, 1;
2, 1, 0, 1;
5, 3, 1, 0, 1;
15, 9, 4, 1, 0, 1;
52, 31, 14, 5, 1, 0, 1;
203, 121, 54, 20, 6, 1, 0, 1;
877, 523, 233, 85, 27, 7, 1, 0, 1;
4140, 2469, 1101, 400, 125, 35, 8, 1, 0, 1;
21147, 12611, 5625, 2046, 635, 175, 44, 9, 1, 0, 1;
MATHEMATICA
t[n_, k_] := t[n, k] = If[k == 0, 1, Sum[t[n-j, k-j] Binomial[n-1, j-1], {j, 1, k}]];
T[n_, k_] := t[n, n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 27 2018 *)
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Feb 10 2011
STATUS
approved