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A143495 Triangle read by rows: 3-Stirling numbers of the second kind. 19
1, 3, 1, 9, 7, 1, 27, 37, 12, 1, 81, 175, 97, 18, 1, 243, 781, 660, 205, 25, 1, 729, 3367, 4081, 1890, 380, 33, 1, 2187, 14197, 23772, 15421, 4550, 644, 42, 1, 6561, 58975, 133057, 116298, 47271, 9702, 1022, 52, 1, 19683, 242461, 724260, 830845, 447195 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

This is the case r = 3 of the r-Stirling numbers of the second kind. The 3-Stirling numbers of the second kind give the number of ways of partitioning the set {1,2,...,n} into k nonempty disjoint subsets with the restriction that the elements 1, 2 and 3 belong to distinct subsets. For remarks on the general case see A143494 (r = 2). The corresponding array of 3-Stirling numbers of the first kind is A143492. The theory of r-Stirling numbers of both kinds is developed in [Broder]. For 3-Lah numbers refer to A143498.

From Peter Bala, Sep 19 2008: (Start)

Let D be the derivative operator d/dx and E the Euler operator x*d/dx. Then x^(-3)*E^n*x^3 = Sum_{k = 0..n} T(n+3,k+3)*x^k*D^k.

The row generating polynomials R_n(x) := Sum_{k= 3..n} T(n,k)*x^k satisfy the recurrence R_(n+1)(x) = x*R_n(x) + x*d/dx(R_n(x)) with R_3(x) = x^3. It follows that the polynomials R_n(x) have only real zeros (apply Corollary 1.2. of [Liu and Wang]).

Relation with the 3-Eulerian numbers E_3(n,j) := A144697(n,j): T(n,k) = 3!/k!*Sum_{j = n-k..n-3} E_3(n,j)*binomial(j,n-k) for n >= k >= 3.

(End)

T(n,k) = S(n,k,3), n>=k>=3, in Mikhailov's first paper, eq.(28) or (A3). E.g.f. column k from (A20) with k->3, r->k. Therefore, with offset [0,0], this triangle is the Sheffer triangle (exp(3*x),exp(x)-1) with e.g.f. of column no. m>=0: exp(3*x)*((exp(x)-1)^m)/m!. See one of the formulas given below. For Sheffer matrices see the W. Lang link under A006232 with the S. Roman reference, also found in A132393. - Wolfdieter Lang, Sep 29 2011

LINKS

Table of n, a(n) for n=3..52.

Broder, Andrei Z., The r-Stirling numbers, Discrete Math. 49, 241-259 (1984)

A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - N. J. A. Sloane, Mar 28 2015]

Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.

V. V. Mikhailov, Ordering of some boson operator functions, J. Phys A: Math. Gen. 16 (1983) 3817-3827.

V. V. Mikhailov, Normal ordering and generalised Stirling numbers, J. Phys A: Math. Gen. 18 (1985) 231-235.

Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001).

L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros [From Peter Bala, Sep 19 2008]

Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.

FORMULA

T(n+3,k+3) = 1/k!*Sum_{i = 0..k} (-1)^(k-i)*C(k,i)*(i+3)^n, n,k >= 0.

T(n,k) = Stirling2(n,k) - 3*Stirling2(n-1,k) + 2*Stirling2(n-2,k), n,k >= 3.

Recurrence relation: T(n,k) = T(n-1,k-1) + k*T(n-1,k) for n > 3, with boundary conditions: T(n,2) = T(2,n) = 0 for all n; T(3,3) = 1; T(3,k) = 0 for k > 3.

Special cases: T(n,3) = 3^(n-3); T(n,4) = 4^(n-3) - 3^(n-3).

E.g.f. (k+3) column (with offset 3): 1/k!*exp(3x)*(exp(x)-1)^k.

O.g.f. k-th column: Sum_{n = k..inf} T(n,k)*x^n = x^k/((1-3*x)*(1-4*x)*...*(1-k*x)).

E.g.f.: exp(3*t + x*(exp(t)-1)) = Sum_{n = 0..inf} Sum_{k = 0..n} T(n+3,k+3)*x^k*t^n/n! = Sum_{n = 0..inf} B_n(3;x)*t^n/n! = 1 + (3+x)*t/1! + (9+7*x+x^2)*t^2/2! + ..., where the row polynomials, B_n(3;x) := Sum_{k = 0..n} T(n+3,k+3)*x^k, may be called the 3- Bell polynomials.

Dobinski-type identities: Row polynomial B_n(3;x) = exp(-x)*Sum_{i = 0..inf} (i+3)^n*x^i/i!; Sum_{k = 0..n} k!*T(n+3,k+3)*x^k = Sum_{i = 0..inf} (i+3)^n*x^i/(1+x)^(i+1).

The T(n,k) are the connection coefficients between the falling factorials and the shifted monomials (x+3)^(n-3). For example, 9 + 7*x + x*(x-1) = (x+3)^2 and 27 + 37*x + 12x*(x-1) + x*(x-1)*(x-2) = (x+3)^3.

This array is the matrix product P^2 * S, where P denotes Pascal's triangle, A007318 and S denotes the lower triangular array of Stirling numbers of the second kind, A008277 (apply Theorem 10 of [Neuwirth]). The inverse array is A049458, the signed 3-Stirling numbers of the first kind.

EXAMPLE

Triangle begins

n\k|....3....4....5....6....7....8

==================================

3..|....1

4..|....3....1

5..|....9....7....1

6..|...27...37...12....1

7..|...81..175...97...18....1

8..|..243..781..660..205...25....1

...

T(5,4) = 7. The set {1,2,3,4,5} can be partitioned into four subsets such that 1, 2 and 3 belong to different subsets in 7 ways: {{1}{2}{3}{4,5}}, {{1}{2}{5}{3,4}}, {{1}{2}{4}{3,5}}, {{1}{3}{4}{2,5}}, {{1}{3}{5}{2,4}}, {{2}{3}{4}{1,5}} and {{2}{3}{5}{1,4}}.

MAPLE

with combinat: T := (n, k) -> (1/(k-3)!)*add ((-1)^(k-i-1)*binomial(k-3, i)*(i+3)^(n-3), i = 0..k-3): for n from 3 to 12 do seq(T(n, k), k = 3..n) end do;

MATHEMATICA

nmax = 12; t[n_, k_] := 1/(k-3)!* Sum[ (-1)^(k-j-1)*Binomial[k-3, j]*(j+3)^(n-3), {j, 0, k-3}]; Flatten[ Table[ t[n, k], {n, 3, nmax}, {k, 3, n}]] (* Jean-François Alcover, Dec 07 2011, after Maple *)

PROG

(Sage)

@CachedFunction

def stirling2r(n, k, r) :

    if n < r: return 0

    if n == r: return 1 if k == r else 0

    return stirling2r(n-1, k-1, r) + k*stirling2r(n-1, k, r)

A143495 = lambda n, k: stirling2r(n, k, 3)

for n in (3..8): [A143495(n, k) for k in (3..n)] # Peter Luschny, Nov 19 2012

CROSSREFS

Cf. A005061 (column 4), A005494 (row sums), A008277, A016753 (column 5), A028025 (column 6), A049458 (matrix inverse), A143492, A143494, A143496, A143498.

Sequence in context: A006803 A197730 A231902 * A245789 A210395 A019770

Adjacent sequences:  A143492 A143493 A143494 * A143496 A143497 A143498

KEYWORD

nonn,easy,tabl

AUTHOR

Peter Bala, Aug 20 2008

STATUS

approved

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Last modified January 18 23:05 EST 2019. Contains 319282 sequences. (Running on oeis4.)