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

0,2

Comments

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

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

Links

Table of n, a(n) for n=0..44.

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.

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.

Example

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).

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 *)

Crossrefs

Cf. A001850, A109980.

Sequence in context: A253583 A130562 A363894 * A154175 A257777 A011208

Adjacent sequences: A152247 A152248 A152249 * A152251 A152252 A152253

Keyword

eigen,nonn,tabl

Author

Gary W. Adamson, Nov 30 2008

Status

approved