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A106828 Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 0 and 0 <= k <= floor(n/2)). 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 (list; graph; refs; listen; history; text; internal format)
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

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)

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-Fran├žois 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

(Sage)

# The function bell_transform is defined in 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

CROSSREFS

See A008306 for more information.

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

Sequence in context: A054877 A269795 A095834 * A055302 A055349 A161174

Adjacent sequences:  A106825 A106826 A106827 * A106829 A106830 A106831

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

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Last modified October 22 03:04 EDT 2019. Contains 328315 sequences. (Running on oeis4.)