A106828 Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 0 and 0 <= k <= floor(n/2)).

1, 0, 0, 1, 0, 2, 0, 6, 3, 0, 24, 20, 0, 120, 130, 15, 0, 720, 924, 210, 0, 5040, 7308, 2380, 105, 0, 40320, 64224, 26432, 2520, 0, 362880, 623376, 303660, 44100, 945, 0, 3628800, 6636960, 3678840, 705320, 34650, 0, 39916800, 76998240, 47324376, 11098780, 866250, 10395

Offset: 0

Author

N. J. A. Sloane, May 22 2005

Keywords

• tabf
• nonn
• easy

Comments

Another version of the triangle is in A008306.

A signed version of this triangle is given by the exponential Riordan array [1, log(1+t)-t]. Its row sums are (-1)^n*(1-n). Another version is [1, log(1-t)+t], whose row sums are 1-n. - Paul Barry, May 10 2008

Examples

			Rows 0 though 7 are:
  1;
  0,
  0,   1;
  0,   2,
  0,   6,   3;
  0,  24,  20,
  0, 120, 130,  15;
  0, 720, 924, 210;
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 256.
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 75.

Links

• Reinhard Zumkeller, Rows n = 0..124 of table, 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 pp. 5, 8.
• Feng-Zhen Zhao, Some Properties of Associated Stirling Numbers, Journal of Integer Sequences, Article 08.1.7, 2008.

Crossrefs

See A008306 for more information.

Cf. A008619 (row lengths), A000166 (row sums).

Programs

Haskell

a106828 n k = a106828_tabf !! n !! k
a106828_row n = a106828_tabf !! n
a106828_tabf = map (fst . fst) $ iterate f (([1], [0]), 1) where
   f ((us, vs), x) =
     ((vs, map (* x) $ zipWith (+) ([0] ++ us) (vs ++ [0])), x + 1)
-- Reinhard Zumkeller, Aug 05 2013

Maple

A106828 := (n,k) -> add(binomial(j,n-2*k)*combinat:-eulerian2(n-k,j),j=0..n-k):
seq(print(seq(A106828(n,k),k=0..iquo(n,2))),n=0..9); # Peter Luschny, Apr 20 2011 (revised Jan 13 2016)
# Second program, after David Callan in A008306:
A106828 := proc(n, k) option remember; if k = 0 then k^n elif k = 1 then (n-1)! elif n <= 2*k-1 then 0 else (n-1)*(procname(n-1, k) + procname(n-2, k-1)) fi end: seq((seq(A106828(n, k), k = 0..iquo(n, 2))), n=0..12); # Peter Luschny, Aug 24 2021

Mathematica

Eulerian2[n_, k_] := Eulerian2[n, k] = If[k == 0, 1, If[k == n, 0, Eulerian2[n - 1, k] (k + 1) + Eulerian2[n - 1, k - 1] (2 n - k - 1)]];
T[n_, k_] := Sum[Binomial[j, n - 2 k] Eulerian2[n - k, j], {j, 0, n - k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n/2}] (* Jean-François Alcover, Jun 13 2019 *)

PARI

E2(n, m) = sum(k=0, n-m, (-1)^(n+k)*binomial(2*n+1, k)*stirling(2*n-m-k+1, n-m-k+1, 1)); \\ A008517
T(n, k) = if ((n==0) && (k==0), 1, sum(j=0, n-k, binomial(j, n-2*k)*E2(n-k, j+1))); \\ Michel Marcus, Dec 07 2021

Python

from math import factorial
def A106828(n, k):
    return k**n if k == 0 else factorial(n-1) if k == 1 else 0 if n <= 2*k - 1 else (n - 1)*(A106828(n-1, k) + A106828(n-2, k-1))
for n in range(14): print([A106828(n, k) for k in range(n//2 + 1)])
# Mélika Tebni, Dec 07 2021, after second Maple script.

Sage

# uses[bell_transform from A264428]
# Computes the full triangle 0<=k<=n.
def A106828_row(n):
    g = lambda k: factorial(k) if k>0 else 0
    s = [g(k) for k in (0..n)]
    return bell_transform(n, s)
[A106828_row(n) for n in (0..8)] # Peter Luschny, Jan 13 2016

Formula

T(n, k) = Sum_{j=0..n-k} binomial(j, n-2*k)*E2(n-k, j), where E2 are the second-order Eulerian numbers (A008517). - Peter Luschny, Jan 13 2016

Also the Bell transform of the sequence g(k) = k! if k>0 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 13 2016

Extensions

Removed extra 0 in row 1 from Michael Somos, Jan 19 2011