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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. 10
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 November 13 10:29 EST 2019. Contains 329093 sequences. (Running on oeis4.)