A387663 Triangle read by rows: T(n,k) = n! * coefficient of m^k in the polynomial counting labeled digraphs with m nodes and n arcs and without directed paths of length >= 2, with 0 <= k <= 2*n.

0, 0, -1, 1, 0, -2, 5, -4, 1, 0, -24, 62, -63, 33, -9, 1, 0, -516, 1522, -1830, 1195, -470, 114, -16, 1, 0, -19440, 61764, -82610, 62380, -29805, 9485, -2040, 290, -25, 1, 0, -1114560, 3756048, -5433032, 4536304, -2464010, 925047, -247502, 47665, -6540, 615

Offset

0,6

Links

Table of n, a(n) for n=0..46.

Formula

T(n,k) = [m^k] n! Sum_{s=0..n} Sum_{t=0..n} binomial(m,s) * binomial(m-s,t) * Sum_{i=0..s} Sum_{j=0..t} (-1)^(i+j) * binomial(s,i) * binomial(t,j) * binomial((s-i)*(t-j), n).

A052296(m,n) = Sum_{k=0..2*n} T(n,k) m^k / n!.

A092552(n) = -Sum_{i=1..n} S2(n,i)*T(i,1).

T(n,k) = Sum_{i=k/2..n} Sum_{s=0..i} Sum_{t=0..i} (-1)^(n-k-i+s+t) * S1(n, i) * S1(s+t, k) * S2(i, s) * S2(i, t), where S1(n,k) = A132393 and S2(n,k) = A008277 are the Stirling numbers of the first and second kind, respectively. - Eytan Chong, Feb 11 2026

Example

Triangle begins:
  0;
  0, -1, 1;
  0, -2, 5, -4, 1;
  0, -24, 62, -63, 33, -9, 1;
  0, -516, 1522, -1830, 1195, -470, 114, -16, 1;
The n = 3 row corresponds to the polynomial (-24*m + 62*m^2 - 63*m^3 + 33*m^4 - 9*m^5 + m^6)/3!, which counts the number of labeled digraphs with m nodes and 3 arcs and without directed paths of length >= 2.

Mathematica

T[n_, k_]:= SeriesCoefficient[ n! Sum[Sum[Binomial[m, s]Binomial[m-s, t] Sum[Sum[ (-1)^(i+j) Binomial[s, i]Binomial[t, j]Binomial[(s-i)(t-j), n], {j, 0, t}], {i, 0, s}], {t, 0, n}], {s, 0, n}], {m, 0, k}]; Table[T[n, k], {n, 5}, {k, 0, 2n}]//Flatten (* Stefano Spezia, Oct 04 2025 *)

Prog

(Python)
import sympy as sp
from sympy.functions.combinatorial import numbers
def T(n, k):
  sum = 0
  for i in range(sp.ceiling(k/2), n+1):
    for s in range(0, i+1):
      for t in range(0, i+1):
        if (s + t >= k):
          sum += numbers.stirling(n, i, signed=True) * numbers.stirling(s+t, k, signed=True) * numbers.stirling(i, s) * numbers.stirling(i, t)
  return sum # Eytan Chong, Feb 12 2026

Crossrefs

Cf. A052296, A092552.

Sequence in context: A217563 A373427 A254881 * A100946 A381848 A377311

Adjacent sequences: A387660 A387661 A387662 * A387664 A387665 A387666

Keyword

tabf,sign

Author

Eytan Chong, Oct 04 2025

Status

approved