A342186 Triangle read by rows, matrix inverse of A139382.

1, -1, 1, 3, -4, 1, -21, 31, -11, 1, 315, -486, 196, -26, 1, -9765, 15381, -6562, 1002, -57, 1, 615195, -978768, 428787, -69688, 4593, -120, 1, -78129765, 124918731, -55434717, 9279163, -652999, 19833, -247, 1

Offset: 1

Author

John Keith, Mar 04 2021

Keywords

• sign
• tabl

Comments

This triangle appears to be the q-analog of A008275 (Stirling numbers of the first kind) for q=2. However, A333142 has a similar definition.

Row sums of unsigned triangle are A006125 with offset 1.

|T(n,k)| is the number of descent free digraphs on [n] containing exactly k source nodes. A descent in a digraph is a pair of vertices s->t such that s>t. A descent free digraph is necessarily acyclic. A source in an acyclic digraph is a node with indegree 0. - Geoffrey Critzer, Mar 05 2025

Examples

			The triangle begins:
           1;
          -1,         1;
           3,        -4,         1;
         -21,        31,       -11,       1;
         315,      -486,       196,     -26,       1;
       -9765,     15381,     -6562,    1002,     -57,     1;
      615195,   -978768,    428787,  -69688,    4593,  -120,    1;
   -78129765, 124918731, -55434717, 9279163, -652999, 19833, -247, 1;
  ...
		

Links

• John Keith, Rows n = 1..20 of triangle, flattened
• Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, arXiv:1909.01550 [math.CO], 20 Mar 2020.

Crossrefs

Cf. A008275, A139382, A333142, A333143, A006125 (row sums).

Columns of unsigned triangle: A005329, A203011, A000295, A203242.

Programs

Maple

A342186 := proc(n, k) if n = 1 and k = 1 then 1 elif k > n or k < 1 then 0 else
A342186(n-1, k-1) - (2^(n-1) - 1) * A342186(n-1, k) fi end:
for n from 1 to 8 do seq(A342186(n, k), k = 1..n) od; # Peter Luschny, Jun 28 2022

Mathematica

T[1, 1] := 1; T[n_, k_] := T[n, k] = If[k > n || k < 1, 0, T[n - 1, k - 1] - (2^(n - 1) - 1)*T[n - 1, k]]; Table[T[n, k], {n, 1, 8}, {k, 1, n}] (* after G. C. Greubel's program for A139382 *)

PARI

mat(nn) = my(m = matrix(nn, nn)); for (n=1, nn, for(k=1, nn, m[n,k] = if (n==1, if (k==1, 1, 0), if (k==1, 1, (2^k-1)*m[n-1, k] + m[n-1, k-1])););); m; \\ A139382
tabl(nn) = 1/mat(nn); \\ Michel Marcus, Mar 18 2021

Formula

T(n,k) = T(n-1,k-1) - (2^(n-1)-1) * T(n-1,k), n, k >= 1, T(1, 1) = 1, T(n, 0) = 0.

For unsigned triangle, T(n, 1) = A005329(n-1); T(n, 2) = A203011(n-1); T(n, n-1) = A000295(n+1); T(n, n-2) = A203242(n-1).

T(n,k) = Sum_{j=k..n} (-1)^(n-j)*2^binomial(n-j,2)*qBinomial(n,j,2)*binomial(j,k), where qBinomial(n,k,2) is A022166(n,k). - Fabian Pereyra, Feb 08 2024