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
Also, the number of distinct k-level fuzzy subsets of a set consisting of n elements ordered by set inclusion. - Rajesh Kumar Mohapatra, Mar 16 2020
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
L. Bartlomiejczyk and J. Drewniak, A characterization of sets and operations invariant under bijections, Aequationes Mathematicae 68 (2004), pp. 1-9.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.
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
Z(P,m) = Sum_{k=0..n} T(n,k)Binomial(m-2,k) = m^n, the zeta polynomial of the poset B_n. Each length m multichain from 0 to 1 in B_n corresponds to a function from [n] into [m]. - Geoffrey Critzer, Dec 25 2020
The entries in row n are the first terms in a table of the successive differences of the sequence [2^n, 3^n, 4^n, ...]. Examples are given below. - Peter Bala, Feb 02 2022
EXAMPLE
Triangle begins
1;
2, 1;
4, 5, 2;
8, 19, 18, 6;
16, 65, 110, 84, 24;
...
From Peter Bala, Feb 02 2022: (Start)
Table of successive differences of k^2 starting at k = 2
4 9 16
5 7
2
gives [4, 5, 2] as row 2 of this triangle.
Table of successive differences of k^3 starting at k = 2
8 27 64 125
19 37 61
18 24
6
gives [8, 19, 8, 6] as row 3 of this triangle. (End)
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-François 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
Diagonal gives A000142. - Rajesh Kumar Mohapatra, Mar 16 2020
KEYWORD
AUTHOR
N. J. A. Sloane, May 02 2000
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 09 2000
STATUS
approved