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A038719 Triangle T(n,k) (0<=k<=n) giving number of chains of length k in partially ordered set formed from subsets of n-set by inclusion. 9
1, 2, 1, 4, 5, 2, 8, 19, 18, 6, 16, 65, 110, 84, 24, 32, 211, 570, 750, 480, 120, 64, 665, 2702, 5460, 5880, 3240, 720, 128, 2059, 12138, 35406, 57120, 52080, 25200, 5040, 256, 6305, 52670, 213444, 484344, 650160, 514080, 221760, 40320, 512, 19171 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The relation of this triangle to A143494 given in the Formula section leads to the following combinatorial interpretation: T(n,k) gives the number of partitions of the set {1,2,...,n+2} into k + 2 blocks where 1 and 2 belong to two distinct blocks and the remaining k blocks are labeled from a fixed set of k labels. - Peter Bala, Jul 10 2014

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

P. Bala, Deformations of the Hadamard product of power series

L. Bartlomiejczyk and J. Drewniak, A characterization of sets and operations invariant under bijections, Aequationes Mathematicae 68 (2004), pp. 1-9.

M. Dukes, C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.

Index entries for sequences related to posets

FORMULA

T(n, k) = Sum_{j=0..k} (-1)^j*C(k, j)*(k+2-j)^n.

T(n+1, k) = k*T(n, k-1) + (k+2)*T(n, k), T(0,0) = 1, T(0,k) = 0 for k>0.

E.g.f.: exp(2*x)/(1+y*(1-exp(x))). - Vladeta Jovovic, Jul 21 2003

A038719 as a lower triangular matrix is the binomial transform of A028246. - Gary W. Adamson, May 15 2005

Binomial transform of n-th row = 2^n + 3^n + 4^n...; e.g., binomial transform of [8, 19, 18, 6] = 2^3 + 3^3 + 4^3 + 5^3... = 8, 27, 64, 125... - Gary W. Adamson, May 15 2005

From Peter Bala, Jul 09 2014: (Start)

T(n,k) = k!*( Stirling2(n+2,k+2) - Stirling2(n+1,k+2) ).

T(n,k) = k!*A143494(n+2,k+2).

n-th row polynomial = 1/(1 + x)*( sum {k >= 0} (k + 2)^n*(x/(1 + x))^k ). Cf. A028246. (End)

The row polynomials have the form (2 + x) o (2 + x) o ... o (2 + x), where o denotes the black diamond multiplication operator of Dukes and White. See example E12 in the Bala link. - Peter Bala, Jan 18 2018

EXAMPLE

Triangle begins

   1;

   2,   1;

   4,   5,   2;

   8,  19,  18,   6;

  16,  65, 110,  84,  24;

  ...

MAPLE

T:= proc(n, k) option remember;

      `if` (n=0, `if`(k=0, 1, 0), k*T(n-1, k-1) +(k+2)*T(n-1, k))

    end:

seq(seq(T(n, k), k=0..n), n=0..10); # Alois P. Heinz, Aug 02 2011

MATHEMATICA

t[n_, k_] := Sum[ (-1)^(k-i)*Binomial[k, i]*(2+i)^n, {i, 0, k}]; Flatten[ Table[ t[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-Fran├žois Alcover, after Pari *)

PROG

(PARI) T(n, k)=sum(i=0, k, (-1)^(k-i)*binomial(k, i)*(2+i)^n)

(Haskell)

a038719 n k = a038719_tabl !! n !! k

a038719_row n = a038719_tabl !! n

a038719_tabl = iterate f [1] where

   f row = zipWith (+) (zipWith (*) [0..] $ [0] ++ row)

                       (zipWith (*) [2..] $ row ++ [0])

-- Reinhard Zumkeller, Jul 08 2012

CROSSREFS

Row sums give A007047. Columns give A000079, A001047, A038721. Next-to-last diagonal gives A038720.

Cf. A028246. A019538, A143494.

Sequence in context: A144332 A209153 A209141 * A125751 A210860 A099492

Adjacent sequences:  A038716 A038717 A038718 * A038720 A038721 A038722

KEYWORD

nonn,easy,nice,tabl

AUTHOR

N. J. A. Sloane, May 02 2000

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000

STATUS

approved

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Last modified October 21 12:36 EDT 2018. Contains 316419 sequences. (Running on oeis4.)