A393523 Triangle T(n,k) read by rows: number of unordered increasing trees on n nodes, with k nodes colored blue and n-k nodes colored red, such that each red node has at least two children.

1, 0, 1, 0, 1, 2, 0, 0, 5, 6, 0, 0, 4, 27, 24, 0, 0, 0, 49, 168, 120, 0, 0, 0, 34, 515, 1200, 720, 0, 0, 0, 0, 790, 5471, 9720, 5040, 0, 0, 0, 0, 496, 13841, 61226, 88200, 40320, 0, 0, 0, 0, 0, 18986, 225687, 730862, 887040, 362880

Offset

1,6

Comments

Also the number of ways of coloring n-k values of {0, ..., n-2} in red, and building a length n-1 inversion sequence in which each red value occurs at least twice. A length n inversion sequence is a sequence of n integers e_1, e_2, ..., e_n such that 0 <= e_i < i.

Links

Benjamin Testart, Table of n, a(n) for n = 1..5050 of rows 1..100

Benjamin Testart, On minimal pattern-containing inversion sequences, arXiv:2602.12130 [math.CO], 2026. See Table 1.

Formula

Mixed (exponential in x and ordinary in y) generating function F(x,y) satisfies d/dx F(x,y) = (y+1)*exp(F(x,y)) - F(x,y) - 1.

Example

Triangle begins:
  [1] 1;
  [2] 0, 1;
  [3] 0, 1, 2;
  [4] 0, 0, 5,  6;
  [5] 0, 0, 4, 27,  24;
  [6] 0, 0, 0, 49, 168,   120;
  [7] 0, 0, 0, 34, 515,  1200,   720;
  [8] 0, 0, 0,  0, 790,  5471,  9720,  5040;
  [9] 0, 0, 0,  0, 496, 13841, 61226, 88200, 40320;

Crossrefs

Main diagonal is A000142.

Row sums are A261001.

Column sums are A393524.

T(2n-1, n) = A002105(n).

Sequence in context: A285192 A278177 A095221 * A078112 A281190 A275619

Adjacent sequences: A393520 A393521 A393522 * A393524 A393525 A393526

Keyword

nonn,tabl

Author

Benjamin Testart, Feb 18 2026

Status

approved