This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A143495 #65 Feb 24 2025 08:52:10
%S A143495 1,3,1,9,7,1,27,37,12,1,81,175,97,18,1,243,781,660,205,25,1,729,3367,
%T A143495 4081,1890,380,33,1,2187,14197,23772,15421,4550,644,42,1,6561,58975,
%U A143495 133057,116298,47271,9702,1022,52,1,19683,242461,724260,830845,447195
%N A143495 Triangle read by rows: 3-Stirling numbers of the second kind.
%C A143495 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.
%C A143495 From _Peter Bala_, Sep 19 2008: (Start)
%C A143495 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.
%C A143495 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]).
%C A143495 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.
%C A143495 (End)
%C A143495 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
%H A143495 Peter Bala, <a href="/A143494/a143494.pdf">Factorising (r,b)-Stirling arrays</a>
%H A143495 Andrei Z. Broder, <a href="http://infolab.stanford.edu/TR/CS-TR-82-949.html">The r-Stirling numbers</a>, Report Number: CS-TR-82-949, Stanford University, Department of Computer Science; see <a href="https://doi.org/10.1016/0012-365X(84)90161-4">also</a>, Discrete Math. 49, 241-259 (1984).
%H A143495 A. Dzhumadildaev and D. Yeliussizov, <a href="http://arxiv.org/abs/1408.6764v1">Path decompositions of digraphs and their applications to Weyl algebra</a>, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
%H A143495 Askar Dzhumadil’daev and Damir Yeliussizov, <a href="https://doi.org/10.37236/5181">Walks, partitions, and normal ordering</a>, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
%H A143495 V. V. Mikhailov, <a href="http://dx.doi.org/10.1088/0305-4470/16/16/019">Ordering of some boson operator functions</a>, J. Phys A: Math. Gen. 16 (1983) 3817-3827.
%H A143495 V. V. Mikhailov, <a href="http://dx.doi.org/10.1088/0305-4470/18/2/012">Normal ordering and generalised Stirling numbers</a>, J. Phys A: Math. Gen. 18 (1985) 231-235.
%H A143495 Erich Neuwirth, <a href="http://homepage.univie.ac.at/erich.neuwirth/papers/TechRep99-05.pdf">Recursively defined combinatorial functions: Extending Galton's board</a>, Discrete Math. 239 No. 1-3, 33-51 (2001).
%H A143495 L. Liu and Y. Wang, <a href="https://arxiv.org/abs/math/0509207">A unified approach to polynomial sequences with only real zeros</a>, arXiv:math/0509207 [math.CO], 2005-2006.
%H A143495 Michael J. Schlosser and Meesue Yoo, <a href="https://doi.org/10.37236/6121">Elliptic Rook and File Numbers</a>, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.
%F A143495 T(n+3,k+3) = (1/k!)*Sum_{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+3)^n, n,k >= 0.
%F A143495 T(n,k) = Stirling2(n,k) - 3*Stirling2(n-1,k) + 2*Stirling2(n-2,k), n,k >= 3.
%F A143495 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.
%F A143495 Special cases: T(n,3) = 3^(n-3); T(n,4) = 4^(n-3) - 3^(n-3).
%F A143495 E.g.f. (k+3) column (with offset 3): (1/k!)*exp(3x)*(exp(x)-1)^k.
%F A143495 O.g.f. k-th column: Sum_{n >= k} T(n,k)*x^n = x^k/((1-3*x)*(1-4*x)*...*(1-k*x)).
%F A143495 E.g.f.: exp(3*t + x*(exp(t)-1)) = Sum_{n >= 0} Sum_{k = 0..n} T(n+3,k+3)*x^k*t^n/n! = Sum_{n >= 0} 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.
%F A143495 Dobinski-type identities: Row polynomial B_n(3;x) = exp(-x)*Sum_{i >= 0} (i+3)^n*x^i/i!; Sum_{k = 0..n} k!*T(n+3,k+3)*x^k = Sum_{i >= 0} (i+3)^n*x^i/(1+x)^(i+1).
%F A143495 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.
%F A143495 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.
%e A143495 Triangle begins
%e A143495 n\k|....3....4....5....6....7....8
%e A143495 ==================================
%e A143495 3..|....1
%e A143495 4..|....3....1
%e A143495 5..|....9....7....1
%e A143495 6..|...27...37...12....1
%e A143495 7..|...81..175...97...18....1
%e A143495 8..|..243..781..660..205...25....1
%e A143495 ...
%e A143495 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}}.
%e A143495 From _Peter Bala_, Feb 23 2025: (Start)
%e A143495 The array factorizes as
%e A143495 / 1 \ /1 \ /1 \ /1 \
%e A143495 | 3 1 | | 3 1 ||0 1 ||0 1 |
%e A143495 | 9 7 1 | = | 9 4 1 ||0 3 1 ||0 0 1 | ...
%e A143495 |27 37 12 1 | |27 13 5 1 ||0 9 4 1 ||0 0 3 1 |
%e A143495 |81 175 97 18 1| |81 40 18 6 1||0 27 13 5 1 ||0 0 9 4 1 |
%e A143495 |... | |... ||... ||... |
%e A143495 where, in the infinite product on the right-hand side, the first array is the Riordan array (1/(1 - 3*x), x/(1 - x)). See A106516. (End)
%p A143495 A143495 := (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(A143495(n, k), k = 3..n) end do;
%t A143495 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 *)
%o A143495 (Sage)
%o A143495 @CachedFunction
%o A143495 def stirling2r(n, k, r) :
%o A143495 if n < r: return 0
%o A143495 if n == r: return 1 if k == r else 0
%o A143495 return stirling2r(n-1, k-1, r) + k*stirling2r(n-1, k, r)
%o A143495 A143495 = lambda n, k: stirling2r(n, k, 3)
%o A143495 for n in (3..8): [A143495(n, k) for k in (3..n)] # _Peter Luschny_, Nov 19 2012
%Y A143495 Cf. A005061 (column 4), A005494 (row sums), A008277, A016753 (column 5), A028025 (column 6), A049458 (matrix inverse), A106516, A143492, A143494, A143496, A143498.
%K A143495 nonn,easy,tabl
%O A143495 3,2
%A A143495 _Peter Bala_, Aug 20 2008