A186020 Eigentriangle of the binomial matrix.

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

Offset: 0

Author

Paul Barry, Feb 10 2011

Keywords

• nonn
• easy
• tabl

Comments

Reversal of Gould triangle A121207. First column is A000110. Second column is A040027.

Row sums are A186021. Diagonal sums are A186022.

Construction is described by Paul D. Hanna in A121207. The method of construction is general for this class of eigentriangle.

Examples

			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;
		

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.

Crossrefs

Cf. A000110, A007318, A121207, A124496, A160185, A186021, A186022.

Programs

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 *)

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