A152250 Eigentriangle, row sums = A001850, the Delannoy numbers.

1, 2, 1, 8, 2, 3, 36, 8, 6, 13, 172, 36, 24, 26, 63, 852, 172, 108, 104, 126, 321, 4324, 852, 516, 468, 504, 642, 1683, 22332, 4324, 2556, 2236, 2268, 2568, 3366, 8989, 116876, 22332, 12972, 11076, 10836, 11556, 13464, 17978, 48639

Offset: 0

Author

Gary W. Adamson, Nov 30 2008

Keywords

• eigen
• nonn
• tabl

Comments

Row sums = A001850, the Delannoy numbers: (1, 3, 13, 63, 321,...).

Sum of n-th row terms = rightmost term of next row.

Examples

			First few rows of the triangle =
  1;
  2, 1;
  8, 2, 3;
  36, 8, 6, 13;
  172, 36, 24, 26, 63;
  852, 172, 108, 104, 126, 321;
  4324, 852, 516, 468, 504, 642, 1683;
  22332, 4324, 2556, 2236, 2268, 2568, 3366, 8989;
  116876, 22332, 12972, 11076, 10836, 11556, 13464, 17978, 48639;
  ...
Row 3 = (36, 8, 6, 13) = termwise products of (36, 8, 2, 1) and (1, 1, 3, 13).
		

Links

• M. Dziemianczuk, Generalizing Delannoy numbers via counting weighted lattice paths, INTEGERS, 13 (2013), #A54.
• M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
• M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, Discrete Mathematics, Volume 339, Issue 3, 6 March 2016, Pages 1116-1139.

Crossrefs

Cf. A001850, A109980.

Programs

Mathematica

nmax = 8;
T[0, 0] = 1;
T[n_, 0] := SeriesCoefficient[1/(x + Sqrt[1 - 6x + x^2]), {x, 0, n}];
T[n_, n_] :=  LegendreP[n - 1, 3];
row[n_] := row[n] = Table[T[m, 0], {m, n, 0, -1}]*Table[T[m, m], {m, 0, n} ];
T[n_, k_] /; 0 < k < n := row[n][[k + 1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

Formula

Triangle read by rows, M*Q. M = an infinite lower triangular matrix with A109980 in every column: (1, 2, 8, 36, 172,...); Q = a matrix with A001850 prefaced with a "1" as the main diagonal: (1, 1, 3, 13, 63, 321,...) and the rest zeros.