A357367 Triangle read by rows. T(n, k) = binomial(n - 1, k - 1)*(n + k)! / k!.

1, 0, 2, 0, 6, 12, 0, 24, 120, 120, 0, 120, 1080, 2520, 1680, 0, 720, 10080, 40320, 60480, 30240, 0, 5040, 100800, 604800, 1512000, 1663200, 665280, 0, 40320, 1088640, 9072000, 33264000, 59875200, 51891840, 17297280

Offset: 0

Author

Peter Luschny, Sep 26 2022

Keywords

• nonn
• tabl

Comments

T(n, k) is the cardinality of the set of all phylogenetic trees with linearly ordered children having n + 1 leaves and k internal vertices. (Proposition 4.16 in Deb and Sokal). - Peter Luschny, Aug 06 2025

Examples

			Triangle T(n, k) starts:
  [0] 1;
  [1] 0,     2;
  [2] 0,     6,      12;
  [3] 0,    24,     120,     120;
  [4] 0,   120,    1080,    2520,     1680;
  [5] 0,   720,   10080,   40320,    60480,    30240;
  [6] 0,  5040,  100800,  604800,  1512000,  1663200,   665280;
  [7] 0, 40320, 1088640, 9072000, 33264000, 59875200, 51891840, 17297280;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows n = 0..150, flattened).
• Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 33.
• Aleks Žigon Tankosič, Recurrence Relations for Some Integer Sequences Related to Ward Numbers, arXiv:2508.04754 [math.CO], 2025. See pp. 3, 5.
• Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See p. 12.

Crossrefs

Cf. A032037 (row sums), A271703, A386789.

Programs

Maple

T := (n, k) -> add((-1)^(m + k) * binomial(n + k, n + m) * binomial(n + m - 1, m - 1) * (n + m)! / m!, m = 0..k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..8);
T := proc(n, k) option remember; if n = 0 and k = 0 then 1 elif k <= 0 or n < 0 then 0 else 2*(n + k - 1)*T(n-1, k-1) + (n + 2*k - 1)*T(n-1, k) fi end:
for n from 0 to 6 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Aug 06 2025

Mathematica

T[n_, k_] := Sum[(-1)^(m + k)*Binomial[n + k, n + m]*Binomial[n + m - 1, m - 1]*(n + m)!/m!, {m, 0, k}]; Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 05 2025 *)

SageMath

def Lah(n, k): return binomial(n, k) * falling_factorial(n - 1, n - k)
def T(n, k): return (sum((-1)^(m + k) * binomial(n + k, n + m) * Lah(n + m, m)
        for m in range(k + 1)))
for n in range(8): print([T(n, k) for k in range(n+1)])

Formula

T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * L(n + m, m), where L denotes the unsigned Lah numbers A271703.

T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * binomial(n + m - 1, m - 1) * (n + m)! / m!.

T(n, k) = (2*(n + k - 1))*T(n-1, k-1) + (n + 2*k - 1)*T(n-1, k) with suitable boundary conditions (from Deb and Sokal). - Peter Luschny, Aug 06 2025

Extensions

New name using a formula of Deb and Sokal by Peter Luschny, Aug 06 2025