OFFSET
0,6
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
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.
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
EXAMPLE
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;
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)
# Alternative: 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 *)
PROG
(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
(SageMath) # 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
(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.
CROSSREFS
KEYWORD
tabf,nonn,easy
AUTHOR
N. J. A. Sloane, May 22 2005
EXTENSIONS
Removed extra 0 in row 1 from Michael Somos, Jan 19 2011
STATUS
approved