A247646 Irregular triangle read by rows, arising in enumeration of lattice paths in strips.

1, 1, 3, 3, 1, 1, 3, 9, 5, 7, 1, 1, 1, 3, 9, 15, 12, 10, 9, 3, 1, 1, 1, 3, 9, 15, 27, 16, 20, 12, 14, 3, 3, 1, 1, 1, 3, 9, 15, 27, 37, 27, 25, 24, 18, 16, 6, 3, 3, 1, 1, 1, 3, 9, 15, 27, 37, 55, 33, 39, 29, 33, 20, 22, 6, 6, 3, 3, 1, 1

Offset

0,3

Comments

It would be nice to have a more explicit definition.

Links

Table of n, a(n) for n=0..69.

Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.

Example

Triangle begins:
  1,
  1,3,3,1,
  1,3,9,5,7,1,1,
  1,3,9,15,12,10,9,3,1,1,
  1,3,9,15,27,16,20,12,14,3,3,1,1,
  ...

Mathematica

v3[x_, k_] := Expand@Cancel[(1+2x+4x^2+2x^3+x^4 - x^(k-1) (k(k+1)/2+2(k+1)x+(-k^2+2k+4)x^2 + (6-2k)x^3 + (6-5k+k^2)x^4/2) + 2x^(2k-1) (k+x-(k-2)x^2) - x^(3k-1)) / ((1-x)^3 (1+x)^2)];
Table[CoefficientList[v3[x, k+2], x], {k, 0, 6}] // Flatten (* Andrei Zabolotskii, Apr 12 2026 *)

Crossrefs

The diagonals on the left are given by A247643.

Sequence in context: A347026 A197928 A109439 * A133333 A296523 A171876

Adjacent sequences: A247643 A247644 A247645 * A247647 A247648 A247649

Keyword

nonn,tabf

Author

N. J. A. Sloane, Sep 23 2014

Extensions

Corrected and extended by Andrei Zabolotskii, Apr 12 2026

Status

approved