A143496 -id:A143496 - cp's OEIS frontend

A008277 Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.

1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1, 1, 31, 90, 65, 15, 1, 1, 63, 301, 350, 140, 21, 1, 1, 127, 966, 1701, 1050, 266, 28, 1, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1, 1, 1023, 28501, 145750, 246730, 179487, 63987, 11880, 1155, 55, 1

Offset: 1

N. J. A. Sloane

Also known as Stirling set numbers and written {n, k}.
S2(n,k) counts partitions of an n-set into k nonempty subsets.
From Manfred Boergens, Apr 07 2025: (Start)
With regard to the preceding comment:
For disjoint collections of subsets see A256894.
For arbitrary collections of subsets see A163353.
For arbitrary collections of nonempty subsets see A055154. (End)
Triangle S2(n,k), 1 <= k <= n, read by rows, given by [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Deléham's operator defined in A084938.
Number of partitions of {1, ..., n+1} into k+1 nonempty subsets of nonconsecutive integers, including the partition 1|2|...|n+1 if n=k. E.g., S2(3,2)=3 since the number of partitions of {1,2,3,4} into three subsets of nonconsecutive integers is 3, i.e., 13|2|4, 14|2|3, 1|24|3. - Augustine O. Munagi, Mar 20 2005
Draw n cards (with replacement) from a deck of k cards. Let prob(n,k) be the probability that each card was drawn at least once. Then prob(n,k) = S2(n,k)*k!/k^n (see A090582). - Rainer Rosenthal, Oct 22 2005
Define f_1(x), f_2(x), ..., such that f_1(x)=e^x and for n = 2, 3, ..., f_{n+1}(x) = (d/dx)(x*f_n(x)). Then f_n(x) = e^x*Sum_{k=1..n} S2(n,k)*x^(k-1). - Milan Janjic, May 30 2008
From Peter Bala, Oct 03 2008: (Start)
For tables of restricted Stirling numbers of the second kind see A143494 - A143496.
S2(n,k) gives the number of 'patterns' of words of length n using k distinct symbols - see [Cooper & Kennedy] for an exact definition of the term 'pattern'. As an example, the words AADCBB and XXEGTT, both of length 6, have the same pattern of letters. The five patterns of words of length 3 are AAA, AAB, ABA, BAA and ABC giving row 3 of this table as (1,3,1).
Equivalently, S2(n,k) gives the number of sequences of positive integers (N_1,...,N_n) of length n, with k distinct entries, such that N_1 = 1 and N_(i+1) <= 1 + max{j = 1..i} N_j for i >= 1 (restricted growth functions). For example, Stirling(4,2) = 7 since the sequences of length 4 having 2 distinct entries that satisfy the conditions are (1,1,1,2), (1,1,2,1), (1,2,1,1), (1,1,2,2), (1,2,2,2), (1,2,2,1) and (1,2,1,2).
(End)
Number of combinations of subsets in the plane. - Mats Granvik, Jan 13 2009
S2(n+1,k+1) is the number of size k collections of pairwise disjoint, nonempty subsets of [n]. For example: S2(4,3)=6 because there are six such collections of subsets of [3] that have cardinality two: {(1)(23)},{(12)(3)}, {(13)(2)}, {(1)(2)}, {(1)(3)}, {(2)(3)}. - Geoffrey Critzer, Apr 06 2009
Consider a set of A000217(n) balls of n colors in which, for each integer k = 1 to n, exactly one color appears in the set a total of k times. (Each ball has exactly one color and is indistinguishable from other balls of the same color.) a(n+1, k+1) equals the number of ways to choose 0 or more balls of each color in such a way that exactly (n-k) colors are chosen at least once, and no two colors are chosen the same positive number of times. - Matthew Vandermast, Nov 22 2010
S2(n,k) is the number of monotonic-labeled forests on n vertices with exactly k rooted trees, each of height one or less. See link "Counting forests with Stirling and Bell numbers" below. - Dennis P. Walsh, Nov 16 2011
If D is the operator d/dx, and E the operator xd/dx, Stirling numbers are given by:  E^n = Sum_{k=1..n} S2(n,k) * x^k*D^k. - Hyunwoo Jang, Dec 13 2011
The Stirling polynomials of the second kind (a.k.a. the Bell / Touchard polynomials) are the umbral compositional inverses of the falling factorials (a.k.a. the Pochhammer symbol or Stirling polynomials of the first kind), i.e., binomial(Bell(.,x),n) = x^n/n! (cf. Copeland's 2007 formulas), implying binomial(xD,n) = binomial(Bell(.,:xD:),n) = :xD:^n/n! where D = d/dx and :xD:^n = x^n*D^n. - Tom Copeland, Apr 17 2014
S2(n,k) is the number of ways to nest n matryoshkas (Russian nesting dolls) so that exactly k matryoshkas are not contained in any other matryoshka. - Carlo Sanna, Oct 17 2015
The row polynomials R(n, x) = Sum_{k=1..n} S2(n, k)*x^k appear in the numerator of the e.g.f. of n-th powers, E(n, x) = Sum_{m>=0} m^n*x^m/m!, as E(n, x) = exp(x)*x*R(n, x), for n >= 1. - Wolfdieter Lang, Apr 02 2017
With offsets 0 for n and k this is the Sheffer product matrix A007318*A048993 denoted by (exp(t), (exp(t) - 1)) with e.g.f. exp(t)*exp(x*(exp(t) - 1)). - Wolfdieter Lang, Jun 20 2017
Number of words on k+1 unlabeled letters of length n+1 with no repeated letters. - Thomas Anton, Mar 14 2019
Also coefficients of moments of Poisson distribution about the origin expressed as polynomials in lambda. [Haight] (see also A331155). - N. J. A. Sloane, Jan 14 2020
k!*S2(n,k) is the number of surjections from an n-element set to a k-element set. - Jianing Song, Jun 01 2022

			The triangle S2(n, k) begins:
\ k    1       2       3        4         5         6         7         8        9
n \   10      11      12       13        14        15       ...
----------------------------------------------------------------------------------
1  |   1
2  |   1       1
3  |   1       3       1
4  |   1       7       6        1
5  |   1      15      25       10         1
6  |   1      31      90       65        15         1
7  |   1      63     301      350       140        21         1
8  |   1     127     966     1701      1050       266        28         1
9  |   1     255    3025     7770      6951      2646       462        36        1
10 |   1     511    9330    34105     42525     22827      5880       750       45
       1
11 |   1    1023   28501   145750    246730    179487     63987     11880     1155
      55       1
12 |   1    2047   86526   611501   1379400   1323652    627396    159027    22275
    1705      66       1
13 |   1    4095  261625  2532530   7508501   9321312   5715424   1899612   359502
   39325    2431      78        1
14 |   1    8191  788970 10391745  40075035  63436373  49329280  20912320  5135130
  752752   66066    3367       91         1
15 |   1   16383 2375101 42355950 210766920 420693273 408741333 216627840 67128490
12662650 1479478  106470     4550       105         1
...
----------------------------------------------------------------------------------
x^4 = 1 x_(1) + 7 x_(2) + 6 x_(3) + 1 x_(4), where x_(k) = P(x,k) = k!*C(x,k). - _Daniel Forgues_, Jan 16 2016

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Index entries for "core" sequences

Cf. A008275 (Stirling numbers of first kind), A048993 (another version of this triangle).
Cf. A000217, A001296, A001297, A001298, A007318, A028246, A039810-A039813, A048994, A087107-A087111, A087127, A094262, A127701.
See also A331155.
Cf. A038207, A086885, A132440, A138378, A173018, A238385.
Cf. A055154, A163353, A256894.
Cf. A000110 (row sums), A102661 (partial row sums).

a008277 n k = a008277_tabl !! (n-1) !! (k-1)
a008277_row n = a008277_tabl !! (n-1)
a008277_tabl = map tail $ a048993_tabl  -- Reinhard Zumkeller, Mar 26 2012

n ((] (1 % !)) * +/@((^~ * (] (1 ^ |.)) * (! {:)@]) i.@>:)) k NB. _Stephen Makdisi, Apr 06 2016

[[StirlingSecond(n,k): k in [1..n]]: n in [1..12]]; // G. C. Greubel, May 22 2019

seq(seq(combinat[stirling2](n, k), k=1..n), n=1..10); # Zerinvary Lajos, Jun 02 2007
stirling_2 := (n,k) -> (1/k!) * add((-1)^(k-i)*binomial(k,i)*i^n, i=0..k);

Table[StirlingS2[n, k], {n, 11}, {k, n}] // Flatten (* Robert G. Wilson v, May 23 2006 *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[1&, rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)
a[n_, n_] := 1; a[n_, 1] := 1;
a[n_, k_] := a[n, k] = a[n-1, k-1] + k a[n-1, k]; Flatten@
Table[a[n, k], {n, 1, 11}, {k, 1, n}] (* Oliver Seipel, Jun 12 2024 *)
With[{m = 11},
 Flatten@MapIndexed[Take[#, #2[[1]]] &,
   Transpose@
    Table[Range[1, m]! Coefficient[(E^x-1)^k/k! + O[x]^(m+1), x,
Range[1, m]], {k, 1, m}]]] (* Oliver Seipel, Jun 12 2024 *)

create_list(stirling2(n+1,k+1),n,0,30,k,0,n); /* Emanuele Munarini, Jun 01 2012 */

for(n=1,22,for(k=1,n,print1(stirling(n,k,2),", "));print()); \\ Joerg Arndt, Apr 21 2013

Stirling2(n,k)=sum(i=0,k,(-1)^i*binomial(k,i)*i^n)*(-1)^k/k!  \\ M. F. Hasler, Mar 06 2012

stirling_number2 # Danny Rorabaugh, Oct 11 2015

S2(n, k) = k*S2(n-1, k) + S2(n-1, k-1), n > 1. S2(1, k) = 0, k > 1. S2(1, 1) = 1.
E.g.f.: A(x, y) = e^(y*e^x-y). E.g.f. for m-th column: (e^x-1)^m/m!.
S2(n, k) = (1/k!) * Sum_{i=0..k} (-1)^(k-i)*binomial(k, i)*i^n.
Row sums: Bell number A000110(n) = Sum_{k=1..n} S2(n, k), n>0.
S(n, k) = Sum (i_1*i_2*...*i_(n-k)) summed over all (n-k)-combinations {i_1, i_2, ..., i_k} with repetitions of the numbers {1, 2, ..., k}. Also S(n, k) = Sum (1^(r_1)*2^(r_2)*...* k^(r_k)) summed over integers r_j >= 0, for j=1..k, with Sum{j=1..k} r_j = n-k. [Charalambides]. - Wolfdieter Lang, Aug 15 2019.
A019538(n, k) = k! * S2(n, k).
A028248(n, k) = (k-1)! * S2(n, k).
For asymptotics see Hsu (1948), among other sources.
Sum_{n>=0} S2(n, k)*x^n = x^k/((1-x)(1-2x)(1-3x)...(1-kx)).
Let P(n) = the number of integer partitions of n (A000041), p(i) = the number of parts of the i-th partition of n, d(i) = the number of distinct parts of the i-th partition of n, p(j, i) = the j-th part of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, and Sum_{i=1..P(n), p(i)=m} = sum running from i=1 to i=P(n) but taking only partitions with p(i)=m parts into account. Then S2(n, m) = Sum_{i=1..P(n), p(i)=m} n!/(Product_{j=1..p(i)} p(i, j)!) * 1/(Product_{j=1..d(i)} m(i, j)!). For example, S2(6, 3) = 90 because n=6 has the following partitions with m=3 parts: (114), (123), (222). Their complexions are: (114): 6!/(1!*1!*4!) * 1/(2!*1!) = 15, (123): 6!/(1!*2!*3!) * 1/(1!*1!*1!) = 60, (222): 6!/(2!*2!*2!) * 1/(3!) = 15. The sum of the complexions is 15+60+15 = 90 = S2(6, 3). - Thomas Wieder, Jun 02 2005
Sum_{k=1..n} k*S2(n,k) = B(n+1)-B(n), where B(q) are the Bell numbers (A000110). - Emeric Deutsch, Nov 01 2006
Recurrence: S2(n+1,k) = Sum_{i=0..n} binomial(n,i)*S2(i,k-1). With the starting conditions S2(n,k) = 1 for n = 0 or k = 1 and S2(n,k) = 0 for k = 0 we have the closely related recurrence S2(n,k) = Sum_{i=k..n} binomial(n-1,i-1)*S2(i-1,k-1). - Thomas Wieder, Jan 27 2007
Representation of Stirling numbers of the second kind S2(n,k), n=1,2,..., k=1,2,...,n, as special values of hypergeometric function of type (n)F(n-1): S2(n,k)= (-1)^(k-1)*hypergeom([ -k+1,2,2,...,2],[1,1,...,1],1)/(k-1)!, i.e., having n parameters in the numerator: one equal to -k+1 and n-1 parameters all equal to 2; and having n-1 parameters in the denominator all equal to 1 and the value of the argument equal to 1. Example: S2(6,k)= seq(evalf((-1)^(k-1)*hypergeom([ -k+1,2,2,2,2,2],[1,1,1,1,1],1)/(k-1)!),k=1..6)=1,31,90,65,15,1. - Karol A. Penson, Mar 28 2007
From Tom Copeland, Oct 10 2007: (Start)
Bell_n(x) = Sum_{j=0..n} S2(n,j) * x^j = Sum_{j=0..n} E(n,j) * Lag(n,-x, j-n) = Sum_{j=0..n} (E(n,j)/n!) * (n!*Lag(n,-x, j-n)) = Sum_{j=0..n} E(n,j) * binomial(Bell.(x)+j, n) umbrally where Bell_n(x) are the Bell / Touchard / exponential polynomials; S2(n,j), the Stirling numbers of the second kind; E(n,j), the Eulerian numbers; and Lag(n,x,m), the associated Laguerre polynomials of order m.
For x = 0, the equation gives Sum_{j=0..n} E(n,j) * binomial(j,n) = 1 for n=0 and 0 for all other n. By substituting the umbral compositional inverse of the Bell polynomials, the lower factorial n!*binomial(y,n), for x in the equation, the Worpitzky identity is obtained; y^n = Sum_{j=0..n} E(n,j) * binomial(y+j,n).
Note that E(n,j)/n! = E(n,j)/(Sum_{k=0..n}E(n,k)). Also (n!*Lag(n, -1, j-n)) is A086885 with a simple combinatorial interpretation in terms of seating arrangements, giving a combinatorial interpretation to the equation for x=1; n!*Bell_n(1) = n!*Sum_{j=0..n} S2(n,j) = Sum_{j=0..n} E(n,j) * (n!*Lag(n, -1, j-n)).
(Appended Sep 16 2020) For connections to the Bernoulli numbers, extensions, proofs, and a clear presentation of the number arrays involved in the identities above, see my post Reciprocity and Umbral Witchcraft. (End)
n-th row = leftmost column of nonzero terms of A127701^(n-1). Also, (n+1)-th row of the triangle = A127701 * n-th row; deleting the zeros. Example: A127701 * [1, 3, 1, 0, 0, 0, ...] = [1, 7, 6, 1, 0, 0, 0, ...]. - Gary W. Adamson, Nov 21 2007
The row polynomials are given by D^n(e^(x*t)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A147315 and A094198. See also A185422. - Peter Bala, Nov 25 2011
Let f(x) = e^(e^x). Then for n >= 1, 1/f(x)*(d/dx)^n(f(x)) = 1/f(x)*(d/dx)^(n-1)(e^x*f(x)) = Sum_{k=1..n} S2(n,k)*e^(k*x). Similar formulas hold for A039755, A105794, A111577, A143494 and A154537. - Peter Bala, Mar 01 2012
S2(n,k) = A048993(n,k), 1 <= k <= n. - Reinhard Zumkeller, Mar 26 2012
O.g.f. for the n-th diagonal is D^n(x), where D is the operator x/(1-x)*d/dx. - Peter Bala, Jul 02 2012
n*i!*S2(n-1,i) = Sum_{j=(i+1)..n} (-1)^(j-i+1)*j!/(j-i)*S2(n,j). - Leonid Bedratyuk, Aug 19 2012
G.f.: (1/Q(0)-1)/(x*y), where Q(k) = 1 - (y+k)*x - (k+1)*y*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 09 2013
From Tom Copeland, Apr 17 2014: (Start)
Multiply each n-th diagonal of the Pascal lower triangular matrix by x^n and designate the result as A007318(x) = P(x).
With Bell(n,x)=B(n,x) defined above, D = d/dx, and :xD:^n = x^n*D^n, a Dobinski formula gives umbrally f(y)^B(.,x) = e^(-x)*e^(f(y)*x). Then f(y)^B(.,:xD:)g(x) = [f(y)^(xD)]g(x) = e^[-(1-f(y)):xD:]g(x) = g[f(y)x].
In particular, for f(y) = (1+y),
A) (1+y)^B(.,x) = e^(-x)*e^((1+y)*x) = e^(x*y) = e^[log(1+y)B(.,x)],
B) (I+dP)^B(.,x) = e^(x*dP) = P(x) = e^[x*(e^M-I)]= e^[M*B(.,x)] with dP = A132440, M = A238385-I = log(I+dP), and I = identity matrix, and
C) (1+dP)^(xD) = e^(dP:xD:) = P(:xD:) = e^[(e^M-I):xD:] = e^[M*xD] with action e^(dP:xD:)g(x) = g[(I+dP)*x].
D) P(x)^m = P(m*x), which implies (Sum_{k=1..m} a_k)^j = B(j,m*x) where the sum is umbrally evaluated only after exponentiation with (a_k)^q = B(.,x)^q = B(q,x). E.g., (a1+a2+a3)^2=a1^2+a2^2+a3^2+2(a1*a2+a1*a3+a2*a3) = 3*B(2,x)+6*B(1,x)^2 = 9x^2+3x = B(2,3x).
E) P(x)^2 = P(2x) = e^[M*B(.,2x)] = A038207(x), the face vectors of the n-Dim hypercubes.
(End)
As a matrix equivalent of some inversions mentioned above, A008277*A008275 = I, the identity matrix, regarded as lower triangular matrices. - Tom Copeland, Apr 26 2014
O.g.f. for the n-th diagonal of the triangle (n = 0,1,2,...): Sum_{k>=0} k^(k+n)*(x*e^(-x))^k/k!. Cf. the generating functions of the diagonals of A039755. Also cf. A112492. - Peter Bala, Jun 22 2014
Floor(1/(-1 + Sum_{n>=k} 1/S2(n,k))) = A034856(k-1), for k>=2. The fractional portion goes to zero at large k. - Richard R. Forberg, Jan 17 2015
From Daniel Forgues, Jan 16 2016: (Start)
Let x_(n), called a factorial term (Boole, 1970) or a factorial polynomial (Elaydi, 2005: p. 60), denote the falling factorial Product_{k=0..n-1} (x-k). Then, for n >= 1, x_(n) = Sum_{k=1..n} A008275(n,k) * x^k, x^n = Sum_{k=1..n} T(n,k) * x_(k), where A008275(n,k) are Stirling numbers of the first kind.
For n >= 1, the row sums yield the exponential numbers (or Bell numbers): Sum_{k=1..n} T(n,k) = A000110(n), and Sum_{k=1..n} (-1)^(n+k) * T(n,k) = (-1)^n * Sum_{k=1..n} (-1)^k * T(n,k) = (-1)^n * A000587(n), where A000587 are the complementary Bell numbers. (End)
Sum_{k=1..n} k*S2(n,k) = A138378(n). - Alois P. Heinz, Jan 07 2022
O.g.f. for the m-th column: x^m/(Product_{j=1..m} 1-j*x). - Daniel Checa, Aug 25 2022
S2(n,k) ~ (k^n)/k!, for fixed k as n->oo. - Daniel Checa, Nov 08 2022
S2(2n+k, n) ~ (2^(2n+k-1/2) * n^(n+k-1/2)) / (sqrt(Pi*(1-c)) * exp(n) * c^n * (2-c)^(n+k)), where c = -LambertW(-2 * exp(-2)). - Miko Labalan, Dec 21 2024
From Mikhail Kurkov, Mar 05 2025: (Start)
For a general proof of the formulas below via generating functions, see Mathematics Stack Exchange link.
Recursion for the n-th row (independently of other rows): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} (j-2)!*binomial(-k,j)*T(n,k+j-1) for 1 <= k < n with T(n,n) = 1 (see Fedor Petrov link).
Recursion for the k-th column (independently of other columns): T(n,k) = 1/(n-k)*Sum_{j=2..n-k+1} binomial(n,j)*T(n-j+1,k)*(-1)^j for 1 <= k < n with T(n,n) = 1. (End)

A143494 Triangle read by rows: 2-Stirling numbers of the second kind.

Original entry on oeis.org

1, 2, 1, 4, 5, 1, 8, 19, 9, 1, 16, 65, 55, 14, 1, 32, 211, 285, 125, 20, 1, 64, 665, 1351, 910, 245, 27, 1, 128, 2059, 6069, 5901, 2380, 434, 35, 1, 256, 6305, 26335, 35574, 20181, 5418, 714, 44, 1, 512, 19171, 111645, 204205, 156660, 58107, 11130, 1110, 54, 1

Offset: 2

Peter Bala, Aug 20 2008

This is the case r = 2 of the r-Stirling numbers of the second kind. The 2-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 and 2 belong to distinct subsets.
More generally, the r-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 numbers 1, 2, ..., r belong to distinct subsets. The case r = 1 gives the usual Stirling numbers of the second kind A008277; for other cases see A143495 (r = 3) and A143496 (r = 4).
The lower unitriangular array of r-Stirling numbers of the second kind equals the matrix product P^(r-1) * S (with suitable offsets in the row and column indexing), where P is Pascal's triangle, A007318 and S is the array of Stirling numbers of the second kind, A008277.
For the definition of and entries relating to the corresponding r-Stirling numbers of the first kind see A143491. For entries on r-Lah numbers refer to A143497. The theory of r-Stirling numbers of both kinds is developed in [Broder].
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^(-2)*E^n*x^2 = Sum_{k = 0..n} T(n+2,k+2)*x^k*D^k.
The row generating polynomials R_n(x) := Sum_{k= 2..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_2(x) = x^2. It follows that the polynomials R_n(x) have only real zeros (apply Corollary 1.2. of [Liu and Wang]).
Relation with the 2-Eulerian numbers E_2(n,j) := A144696(n,j): T(n,k) = 2!/k!*Sum_ {j = n-k..n-2} E_2(n,j)*binomial(j,n-k) for n >= k >= 2. (End)
From Wolfdieter Lang, Sep 29 2011: (Start)
T(n,k) = S(n,k,2), n>=k>=2, in Mikhailov's first paper, eq.(28) or (A3). E.g.f. column no. k from (A20) with k->2, r->k. Therefore, with offset [0,0], this triangle is the Sheffer triangle (exp(2*x),exp(x)-1) with e.g.f. of column no. m>=0: exp(2*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. (End)

			Triangle begins
  n\k|...2....3....4....5....6....7
  =================================
  2..|...1
  3..|...2....1
  4..|...4....5....1
  5..|...8...19....9....1
  6..|..16...65...55...14....1
  7..|..32..211..285..125...20....1
  ...
T(4,3) = 5. The set {1,2,3,4} can be partitioned into three subsets such that 1 and 2 belong to different subsets in 5 ways: {{1}{2}{3,4}}, {{1}{3}{2,4}}, {{1}{4}{2,3}}, {{2}{3}{1,4}} and {{2}{4}{1,3}}; the remaining possibility {{1,2}{3}{4}} is not allowed.
From _Peter Bala_, Feb 23 2025: (Start)
The array factorizes as
/ 1               \       /1             \ /1             \ /1            \
| 2    1           |     | 2   1          ||0  1           ||0  1          |
| 4    5   1       |  =  | 4   3   1      ||0  2   1       ||0  0  1       | ...
| 8   19   9   1   |     | 8   7   4  1   ||0  4   3  1    ||0  0  2  1    |
|16   65  55  14  1|     |16  15  11  6  1||0  8   7  4  1 ||0  0  4  3  1 |
|...               |     |...             ||...            ||...           |
where, in the infinite product on the right-hand side, the first array is the Riordan array (1/(1 - 2*x), x/(1 - x)). See A055248. (End)

Peter Bala, Factorising (r,b)-Stirling arrays
S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, Unit interval parking functions and the r-Fubini numbers, arXiv:2401.06937 [math.CO], 2024. See page 2.
Andrei Z. Broder, The r-Stirling numbers, Report Number: CS-TR-82-949, 1982, Stanford University, Department of Computer Science.
Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
C. B. Corcino, L. C. Hsu, and E. L. Tan, Asymptotic approximations of r-Stirling numbers, Approximation Theory Appl. 15, No. 3 13-25 (1999).
A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764 [math.CO], 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.
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See p. 8.
L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.
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, Technical Report TR 99-05, July 1999, Universität Wien.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001).
M. d’Ocagne, Sur une classe de nombres remarquables, Amer. J. Math., Vol. 9 (1887), 353-380.
Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.
Mark Shattuck, Generalized r-Lah numbers, arXiv:1412.8721 [math.CO], 2014.

A001047 (column 3), A005493 (row sums), A008277, A016269 (column 4), A025211 (column 5), A049444 (matrix inverse), A074051 (alt. row sums).

Cf. A055248, A143491, A143495, A143496, A143497.

with combinat: T := (n, k) -> (1/(k-2)!)*add ((-1)^(k-i)*binomial(k-2,i)*(i+2)^(n-2),i = 0..k-2): for n from 2 to 11 do seq(T(n, k), k = 2..n) end do;

t[n_, k_] := StirlingS2[n, k] - StirlingS2[n-1, k]; Flatten[ Table[ t[n, k], {n, 2, 11}, {k, 2, n}]] (* Jean-François Alcover, Dec 02 2011 *)

@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)
A143494 = lambda n,k: stirling2r(n, k, 2)
for n in (2..6):
    [A143494(n, k) for k in (2..n)] # Peter Luschny, Nov 19 2012

T(n+2,k+2) = (1/k!)*Sum_{i = 0..k} (-1)^(k-i)*C(k,i)*(i+2)^n, n,k >= 0.
T(n,k) = Stirling2(n,k) - Stirling2(n-1,k) for n, k >= 2.
Recurrence relation: T(n,k) = T(n-1,k-1) + k*T(n-1,k) for n > 2, with boundary conditions T(n,1) = T(1,n) = 0 for all n, T(2,2) = 1 and T(2,k) = 0 for k > 2. Special cases: T(n,2) = 2^(n-2); T(n,3) = 3^(n-2) - 2^(n-2).
As a sum of monomial functions of degree m: T(n+m,n) = Sum_{2 <= i_1 <= ... <= i_m <= n} (i_1*i_2*...*i_m). For example, T(6,4) = Sum_{2 <= i <= j <= 4} (i*j) = 2*2 + 2*3 + 2*4 + 3*3 + 3*4 + 4*4 = 55.
E.g.f. column k+2 (with offset 2): 1/k!*exp(2*x)*(exp(x) - 1)^k.
O.g.f. k-th column: Sum_{n >= k} T(n,k)*x^n = x^k/((1-2*x)*(1-3*x)*...*(1-k*x)).
E.g.f.: exp(2*t + x*(exp(t) - 1)) = Sum_{n >= 0} Sum_{k = 0..n} T(n+2,k+2) *x^k*t^n/n! = Sum_{n >= 0} B_n(2;x)*t^n/n! = 1 + (2 + x)*t/1! + (4 + 5*x + x^2)*t^2/2! + ..., where the row polynomial B_n(2;x) := Sum_{k = 0..n} T(n+2,k+2)*x^k denotes the 2-Bell polynomial.
Dobinski-type identities: Row polynomial B_n(2;x) = exp(-x)*Sum_{i >= 0} (i + 2)^n*x^i/i!. Sum_{k = 0..n} k!*T(n+2,k+2)*x^k = Sum_{i >= 0} (i + 2)^n*x^i/(1 + x)^(i+1).
The T(n,k) are the connection coefficients between falling factorials and the shifted monomials (x + 2)^(n-2). For example, from row 4 we have 4 + 5*x + x*(x - 1) = (x + 2)^2, while from row 5 we have 8 + 19*x + 9*x*(x - 1) + x*(x - 1)*(x - 2) = (x + 2)^3.
The row sums of the array are the 2-Bell numbers, B_n(2;1), equal to A005493(n-2). The alternating row sums are the complementary 2-Bell numbers, B_n(2;-1), equal to (-1)^n*A074051(n-2).
This array is the matrix product P * S, where P denotes the Pascal triangle, A007318 and S denotes the lower triangular array of Stirling numbers of the second kind, A008277 (apply Theorem 10 of [Neuwirth]).
Also, this array equals the transpose of the upper triangular array A126351. The inverse array is A049444, the signed 2-Stirling numbers of the first kind. See A143491 for the unsigned version of the inverse.
Let f(x) = exp(exp(x)). Then for n >= 1, the row polynomials R(n,x) are given by R(n+2,exp(x)) = 1/f(x)*(d/dx)^n(exp(2*x)*f(x)). Similar formulas hold for A008277, A039755, A105794, A111577 and A154537. - Peter Bala, Mar 01 2012

A143495 Triangle read by rows: 3-Stirling numbers of the second kind.

Original entry on oeis.org

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

Offset: 3

Peter Bala, Aug 20 2008

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

			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}}.
From _Peter Bala_, Feb 23 2025: (Start)
The array factorizes as
/ 1               \       /1             \ /1             \ /1            \
| 3    1           |     | 3   1          ||0  1           ||0  1          |
| 9    7   1       |  =  | 9   4   1      ||0  3   1       ||0  0  1       | ...
|27   37  12   1   |     |27  13   5  1   ||0  9   4  1    ||0  0  3  1    |
|81  175  97  18  1|     |81  40  18  6  1||0 27  13  5  1 ||0  0  9  4  1 |
|...               |     |...             ||...            ||...           |
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)

Peter Bala, Factorising (r,b)-Stirling arrays
Andrei Z. Broder, The r-Stirling numbers, Report Number: CS-TR-82-949, Stanford University, Department of Computer Science; see also, 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 [math.CO], 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 and Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.
Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.

Cf. A005061 (column 4), A005494 (row sums), A008277, A016753 (column 5), A028025 (column 6), A049458 (matrix inverse), A106516, A143492, A143494, A143496, A143498.

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;

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 *)

@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

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

A193685 5-Stirling numbers of the second kind.

Original entry on oeis.org

1, 5, 1, 25, 11, 1, 125, 91, 18, 1, 625, 671, 217, 26, 1, 3125, 4651, 2190, 425, 35, 1, 15625, 31031, 19981, 5590, 740, 45, 1, 78125, 201811, 170898, 64701, 12250, 1190, 56, 1, 390625, 1288991, 1398097, 688506, 174951, 24150, 1806, 68, 1, 1953125, 8124571, 11075670, 6906145, 2263065, 416451, 44016, 2622, 81, 1, 9765625, 50700551, 85654261, 66324830, 27273730, 6427575, 900627, 75480, 3675, 95, 1

Offset: 0

Wolfdieter Lang, Oct 06 2011

This is the lower triangular Sheffer matrix (exp(5*x),exp(x)-1). For Sheffer matrices see the W. Lang link under A006232 with references, and the rules for the conversion to the umbral notation of S. Roman's book.
The general case is Sheffer (exp(r*x),exp(x)-1), r=0,1,..., corresponding to r-Stirling2 numbers with row and column offsets 0. See the Broder link for r-Stirling2 numbers with offset [r,r].
a(n,m), n >= m >= 0, gives the number of partitions of the set {1.2....,n+5} into m+5 nonempty distinct subsets such that 1,2,3,4 and 5 belong to distinct subsets.
a(n,m) appears in the following normal ordering of Bose operators a and a* satisfying the Lie algebra [a,a*]=1: (a*a)^n (a*)^5 = Sum_{m=0..n} a(n,m)*(a*)^(5+m)*a^m, n >= 0. See the Mikhailov papers, where a(n,m) = S(n+5,m+5,5).
  With a->D=d/dx and a*->x we also have
  (xD)^n x^5 = Sum_{m=0..n} a(n,m)*x^(5+m)*D^m, n >= 0.

			n\m  0    1    2   3  4  5 ...
0    1
1    5    1
2   25   11    1
3  125   91   18   1
4  625  671  217  26  1
5 3125 4651 2190 425 35  1
...
5-restricted S2: a(1,0)=5 from 1,6|2|3|4|5, 2,6|1|3|4|5,
3,6|1|2|4|5, 4,6|1|2|3|5 and 5,6|1|2|3|4.
Recurrence: a(4,2) = (5+2)*a(3,2)+ a(3,1) = 7*18 + 91 = 217.
Normal ordering (n=1): (xD)^1 x^5 = Sum_{m=0..1} a(1,m)*x^(5+m)*D^m = 5*x^5 + 1*x^6*D.
a(2,1) = Sum_{j=0..1} S1(5,5-j)*S2(7-j,6) = 1*21 - 10*1 = 11.

Vincenzo Librandi, Rows n = 0..100, flattened
Peter Bala, Generalized Dobinski formulas
Andrei Z. Broder, 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 [math.CO], 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.

Cf. A048993, A143494, A143495, A143496.
Cf. A196834 (row sums), A196835 (alternating row sums).
Columns: A000351 (m=0), A005062 (m=1), A019757 (m=2), A028165 (m=3), ...

a[n_, m_] := Sum[ StirlingS1[5, 5-j]*StirlingS2[n+5-j, m+5], {j, 0, Min[5, n-m]}]; Flatten[ Table[ a[n, m], {n, 0, 10}, {m, 0, n}] ] (* Jean-François Alcover, Dec 02 2011, after Wolfdieter Lang *)

E.g.f. of row polynomials s(n,x):=Sum_{m=0..n} a(n,m)*x^m: exp(5*z + x(exp(z)-1)).
E.g.f. of column no. m (with leading zeros):
  exp(5*x)*((exp(x)-1)^m)/m!, m >= 0 (Sheffer).
O.g.f. of column no. m (without leading zeros):
  1/Product_{j=0..m} (1-(5+j)*x), m >= 0. (Compute the first derivative of the column e.g.f. and compare its Laplace transform with the partial fraction decomposition of the o.g.f. x^(m-1)/Product_{j=0..m} (1-(5+j)*x). This works for every r-restricted Stirling2 triangle.)
Recurrence: a(n,m) = (5+m)*a(n-1,m) + a(n-1,m-1), a(0,0)=1, a(n,m)=0 if n < m, a(n,-1)=0.
a(n,m) = Sum_{j=0..min(5,n-m)} S1(5,5-j)*S2(n+5-j,m+5), n >= m >= 0, with S1 and S2 the Stirling1 and Stirling2 numbers A008275 and A048993, respectively (see the Mikailov papers).
Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k>=0} k*(4+k)^(n-1)*x^(k-1)/k!. - Peter Bala, Jun 23 2014

A003468 Number of minimal 3-covers of a labeled n-set.

Original entry on oeis.org

1, 22, 305, 3410, 33621, 305382, 2619625, 21554170, 171870941, 1337764142, 10216988145, 76862115330, 571247591461, 4203844925302, 30687029023865, 222518183370890, 1604626924403181, 11518132293452862

Offset: 3

N. J. A. Sloane

nonn

This is also the fourth column of the Sheffer triangle A143496 (4-restricted Stirling2 numbers). See the e.g.f. given below. See also the Sheffer comments in A193685. - Wolfdieter Lang, Oct 08 2011

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 3..1000
T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Minimal cover.
Index entries for linear recurrences with constant coefficients, signature (22, -179, 638, -840).

Cf. A000392, A003468, A016111, A046166-A046169, A057668, A005783-A005786, A055066.

Cf. A143496, A000302, A005060, A016103.

[7^n/6 - 6^n/2 + 5^n/2 - 4^n/6: n in [3..30]]; // Vincenzo Librandi, May 03 2013

A003468:=1/(6*z-1)/(4*z-1)/(7*z-1)/(5*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

Table[7^n/6 - 6^n/2 + 5^n/2 - 4^n/6, {n, 3, 20}] (* Vaclav Kotesovec, Nov 19 2012 *)
LinearRecurrence[{22,-179,638,-840},{1,22,305,3410},20] (* Harvey P. Dale, Jan 09 2024 *)

G.f.: x^3/((1 - 4*x)*(1 - 5*x)*(1 - 6*x)*(1 - 7*x)). - N. J. A. Sloane, May 12 1994, corrected by Vaclav Kotesovec, Nov 19 2012
E.g.f.: (exp(4*x)*(exp(x) - 1)^3)/6. More generally, e.g.f. for number of minimal m-covers of a labeled n-set is (exp((2^m - m - 1)*x)*(exp(x) - 1)^m)/m!. - Vladeta Jovovic, May 09 2004
If we define f(m, j, x) = sum(binomial(m, k)*stirling2(k, j)*x^(m - k),k = j .. m) then a(n) = f(n, 3, 4), (n >= 3). - Milan Janjic, Apr 26 2009
a(n) = 7^n/6 - 6^n/2 + 5^n/2 - 4^n/6. - Vaclav Kotesovec, Nov 19 2012

A045379 Expansion of e.g.f.: exp(4*z + exp(z) - 1).

Original entry on oeis.org

1, 5, 26, 141, 799, 4736, 29371, 190497, 1291020, 9131275, 67310847, 516369838, 4116416797, 34051164985, 291871399682, 2588914083065, 23733360653955, 224592570163192, 2191466128865567, 22024934452712437, 227771488390279260

Offset: 0

N. J. A. Sloane

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500
R. Jakimczuk, Successive Derivatives and Integer Sequences, J. Int. Seq. 14 (2011) # 11.7.3.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
I. Mezo, The r-Bell numbers, J. Int. Seq. 14 (2011) # 11.1.1.

Cf. A000110, A005493, A005494, A108087, A196834.

Equals the row sums of triangle A143496. - Wolfdieter Lang, Sep 29 2011

A045379:= func< n | (&+[Binomial(n,j)*4^(n-j)*Bell(j): j in [0..n]]) >;
[A045379(n): n in [0..30]]; // G. C. Greubel, Dec 01 2022

a[0]= 1; a[n_]:= a[n]= 4*a[n-1] +Sum[Binomial[n-1, k]*a[k], {k,0,n-1}]; Array[a, 21, 0] (* Amiram Eldar, Jul 03 2020 *)

def A045379(n): return sum( 4^(n-j)*bell_number(j)*binomial(n,j) for j in range(n+1))
[A045379(n) for n in range(31)] # G. C. Greubel, Dec 01 2022

a(n) = exp(-1)*Sum_{k>=0} ((k+4)^n)/k!. - Gerald McGarvey, Jun 03 2004
A recursive formula to compute some integer sequences (including A000110, A005493, A005494 and the present sequence). Define G(n, m), where n, m >= 0, as follows: G(0, m) = 1; G(n, m) = G(n-1, m) * (m+1) + G(n-1, m+1), where n > 0. Then G(n, 0) = A000110(n+1); G(n, 1) = A005493(n+1); G(n, 2) = A005494(n+1); G(n, 3) = A045379(n+1). - Alexey Andreev (ava12(AT)nm.ru), Jan 05 2006
Define f_1(x), f_2(x), ... such that f_1(x)=x^3*e^x, f_{n+1}(x) = (d/dx)(x*f_n(x)), for n=2,3,.... Then a(n-1) = e^(-1)*f_n(1). - Milan Janjic, May 30 2008
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i <= j), and A[i,j]=0 otherwise. Then, for n >= 1, a(n) = (-1)^(n)*charpoly(A,-4). - Milan Janjic, Jul 08 2010
G.f.: 1/U(0) where U(k) = 1 - x*(k+5) - x^2*(k+1)/U(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 4) / LambertW(n)^(n + 9/2). - Vaclav Kotesovec, Jun 10 2020
a(0) = 1; a(n) = 4 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 02 2020
a(n) = Sum_{j=0..n} binomial(n, j)*4^(n-j)*A000110(j). - G. C. Greubel, Dec 01 2022

A049459 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -4, 1, 20, -9, 1, -120, 74, -15, 1, 840, -638, 179, -22, 1, -6720, 5944, -2070, 355, -30, 1, 60480, -60216, 24574, -5265, 625, -39, 1, -604800, 662640, -305956, 77224, -11515, 1015, -49, 1, 6652800, -7893840, 4028156, -1155420, 203889

Offset: 0

Wolfdieter Lang

a(n,m)= ^4P_n^m in the notation of the given reference with a(0,0) := 1.
The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(4+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1.
In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(4*t),exp(t)-1).
See A143493 for the unsigned version of this array and A143496 for the inverse. - Peter Bala, Aug 25 2008

			   1;
  -4,    1;
  20,   -9,   1;
-120,   74, -15,   1;
840, -638, 179, -22, 1;

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Unsigned column sequences are: A001715-A001719. Cf. A008275 (Stirling1 triangle), A049458, A049460. Row sums (signed triangle): A001710(n+2)*(-1)^n. Row sums (unsigned triangle): A001720(n+4).
Cf. A000035 A084938.
A143493, A143496. - Peter Bala, Aug 25 2008

a049459 n k = a049459_tabl !! n !! k
a049459_row n = a049459_tabl !! n
a049459_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 4)
-- Reinhard Zumkeller, Mar 11 2014

A049459_row := n -> seq((-1)^(n-k)*coeff(expand(pochhammer(x+4, n)), x, k), k=0..n): seq(print(A049459_row(n)),n=0..8); # Peter Luschny, May 16 2013

a[n_, m_] /; 0 <= m <= n := a[n, m] = a[n-1, m-1] - (n+3)*a[n-1, m];
a[n_, m_] /; n < m = 0;
a[_, -1] = 0; a[0, 0] = 1;
Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)

a(n, m)= a(n-1, m-1) - (n+3)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n

Triangle (signed) = [ -4, -1, -5, -2, -6, -3, -7, -4, -8, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [4, 1, 5, 2, 6, 3, 7, 4, 8, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]; where DELTA is Deléham's operator defined in A084938 (unsigned version in A143493).

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,4), for n=1,2,...;i=0...n. - Milan Janjic, Dec 21 2008

Second formula corrected by Philippe Deléham, Nov 09 2008

A143497 Triangle of unsigned 2-Lah numbers.

Original entry on oeis.org

1, 4, 1, 20, 10, 1, 120, 90, 18, 1, 840, 840, 252, 28, 1, 6720, 8400, 3360, 560, 40, 1, 60480, 90720, 45360, 10080, 1080, 54, 1, 604800, 1058400, 635040, 176400, 25200, 1890, 70, 1, 6652800, 13305600, 9313920, 3104640, 554400, 55440, 3080, 88, 1

Offset: 2

Peter Bala, Aug 25 2008

For a signed version of this triangle see A062137. The unsigned 2-Lah number L(2; n,k) gives the number of partitions of the set {1, 2, ..., n} into k ordered lists with the restriction that the elements 1 and 2 must belong to different lists. More generally, the unsigned r-Lah number L(r; n, k) gives the number of partitions of the set {1, 2, ..., n} into k ordered lists with the restriction that the elements 1, 2, ..., r belong to different lists. If r = 1 there is no restriction and we obtain the unsigned Lah numbers A105278. For other cases see A143498 (r=3) and A143499 (r=4). We make some remarks on the general case.
The unsigned r-Lah numbers occur as connection constants in the generalized Lah identity (x + 2*r - 1)*(x + 2*r)*...*(x + 2*r + n - r - 2) = Sum_{k=r..n} L(r; n, k)*(x - 1)*(x - 2)*...*(x - k + r) for n >= r and where any empty products are taken equal to 1 (for a bijective proof of the identity, follow the proof of [Petkovsek and Pisanski] but restrict the first r of the Argonauts to different paths).
The unsigned r-Lah numbers satisfy the same recurrence as the unsigned Lah numbers, namely, L(r; n, k) = (n + k - 1)*L(r; n - 1,k) + L(r; n - 1,k - 1), but with the boundary conditions: L(r; n, k) = 0 if n < r or if k < r; L(r; r, r) = 1. The recurrence has the explicit solution L(r; n, k) = ((n - r)!/(k - r)!)*binomial(n + r - 1, k + r - 1) for n, k >= r. It follows that the unsigned r-Lah numbers have 'vertical' generating functions for k >= r of the form Sum_{n>=k} L(r; n, k)*t^n/(n -r)! = 1/(k - r)!*t^k/(1 - t)^(k + r). This yields the e.g.f. for the array of unsigned r-restricted Lah numbers in the form: Sum_{n,k>=r} L(r; n, k)*x^k*t^n/(n-r)! = (x*t)^r * 1/(1 - t)^(2*r) * exp(x*t/(1 - t)) = (x*t)^r (1 + (2*r + x)*t + (2r*(2*r + 1) + 2*(2*r + 1)*x + x^2)*t^2/2! + ...).
The array of unsigned r-Lah numbers begins
  1
  2r                  1
  2r*(2r+1)           2*(2r+1)           1
  2r*(2r+1)*(2r+2)    3*(2r+1)*(2r+2)    3*(2r+2)      1
 ...
and equals exp(D(r)), where D(r) is the array with the sequence (2*r, 2*(2*r + 1), 3*(2*r + 2), 4*(2*r + 3), ...) on the main subdiagonal and zeros everywhere else.
The unsigned r-Lah numbers are related to the r-Stirling numbers: the lower triangular array of unsigned r-Lah numbers may be expressed as the matrix product St1(r) * St2(r), where St1(r) and St2(r) denote the arrays of r-Stirling numbers of the first and second kind respectively. The theory of r-Stirling numbers is developed in [Broder]. See A143491 - A143496 for tables of r-Stirling numbers. An alternative factorization for the array is as St1 * P^(2r - 2) * St2, where P denotes Pascal's triangle, A007318, St1 is the triangle of unsigned Stirling numbers of the first kind, abs(A008275) and St2 denotes the triangle of Stirling numbers of the second kind, A008277 (apply Theorem 10 of [Neuwirth]).
The array of unsigned r-Lah numbers is an example of the fundamental matrices sketched in A133314. So redefining the offset as n=0, given matrices A and B with A(n, k) = T(n, k)*a(n - k) and B(n, k) = T(n, k)*b(n - k), then A*B = C where C(n, k) = T(n,k)*[a(.) + b(.)]^(n - k), umbrally. An e.g.f. for the row polynomials of A is exp(x*t) exp{-x*t*[a*t/(a*t - 1)]}/(1 - a*t)^4 = exp(x*t) exp[(.)!*Laguerre(., 3, -x*t)* a(.)*t)], umbrally. - Tom Copeland, Sep 19 2008

			Triangle begins:
=========================================
n\k |     2     3     4     5     6     7
----+------------------------------------
  2 |     1
  3 |     4     1
  4 |    20    10     1
  5 |   120    90    18     1
  6 |   840   840   252    28     1
  7 |  6720  8400  3360   560    40     1
 ...
T(4,3) = 10. The ten partitions of {1,2,3,4} into 3 ordered lists such that the elements 1 and 2 lie in different lists are: {1}{2}{3,4} and {1}{2}{4,3}, {1}{3}{2,4} and {1}{3}{4,2}, {1}{4}{2,3} and {1}{4}{3,2}, {2}{3}{1,4} and {2}{3}{4,1}, {2}{4}{1,3} and {2}{4}{3,1}. The remaining two partitions {3}{4}{1,2} and {3}{4}{2,1} are not allowed because the elements 1 and 2 belong to the same block.

Muniru A Asiru, Table of n, a(n) for n = 2..4951 Rows n = 2..100
A. Z. Broder, The r-Stirling numbers, Report CS-TR-82-949, Stanford University, Department of Computer Science, 1982.
A. Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
Gi-Sang Cheon and Ji-Hwan Jung, r-Whitney numbers of Dowling lattices, Discrete Math., 312 (2012), 2337-2348.
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Tech Report TR 99-05, 1999.
Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001).
G. Nyul and G. Rácz, The r-Lah numbers, Discrete Mathematics, 338 (2015), 1660-1666.
Marko Petkovsek and Tomaz Pisanski, Combinatorial interpretation of unsigned Stirling and Lah numbers, University of Ljubljana, Preprint series, Vol. 40 (2002), 837.
Jose L. Ramirez and M. Shattuck, A (p, q)-Analogue of the r-Whitney-Lah Numbers, Journal of Integer Sequences, 19, 2016, #16.5.6.
Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31.
M. Shattuck, Generalized r-Lah numbers, arXiv:1412.8721 [math.CO], 2014.

Cf. A001715 (column 2), A007318, A008275, A008277, A061206 (column 3), A062137, A062141 - A062144 ( column 4 to column 7), A062146 (alt. row sums), A062147 (row sums), A105278 (unsigned Lah numbers), A143491, A143494, A143498, A143499.

T:=Flat(List([2..10],n->List([2..n],k->(Factorial(n-2)/Factorial(k-2))*Binomial(n+1,k+1)))); # Muniru A Asiru, Nov 27 2018

T := (n, k) -> ((n-2)!/(k-2)!)*binomial(n+1, k+1):
for n from 2 to 11 do seq(T(n, k), k = 2..n) od;

T[n_, k_] := (n-2)!/(k-2)!*Binomial[n+1, k+1]; Table[T[n, k], {n,2,10}, {k,2,n}] // Flatten (* Amiram Eldar, Nov 27 2018 *)

create_list((n - 2)!/(k - 2)!*binomial(n + 1, k + 1), n, 2, 12, k, 2, n); /* Franck Maminirina Ramaharo, Nov 27 2018 */

T(n, k) = ((n - 2)!/(k - 2)!)*C(n+1, k+1), for n, k >= 2.
Recurrence: T(n, k) = (n + k - 1)*T(n-1, k) + T(n-1, k-1) for n, k >= 2, with the boundary conditions: T(n, k) = 0 if n < 2 or k < 2; T(2, 2) = 1.
E.g.f. for column k: Sum_{n>=k} T(n, k)*t^n/(n - 2)! = 1/(k - 2)!*t^k/(1 - t)^(k+2) for k >= 2.
E.g.f: Sum_{n=2..inf} Sum_{k=2..n} T(n, k)*x^k*t^n/(n - 2)! = (x*t)^2/(1 - t)^4* exp(x*t/(1 - t)) = (x*t)^2*(1 + (4 + x)*t + (20 + 10*x + x^2)*t^2/2! + ... ).
Generalized Lah identity: (x + 3)*(x + 4)*...*(x + n) = Sum_{k = 2..n} T(n, k)*(x - 1)*(x - 2)*...*(x - k + 2).
The polynomials 1/n!*Sum_{k=2..n+2} T(n+2, k)*(-x)^(k - 2) for n >= 0 are the generalized Laguerre polynomials Laguerre(n,3,x). See A062137.
Array = A143491 * A143494 = abs(A008275) * (A007318)^2 * A008277 (apply Theorem 10 of [Neuwirth]). Array equals exp(D), where D is the array with the quadratic sequence (4, 10, 18, 28, ...) on the main subdiagonal and zeros elsewhere.
After adding 1 to the head of the main diagonal and a zero to each of the subdiagonals, the n-th diagonal may be generated as coefficients of (1/n!) [D^(-1) tDt t^(-3)D t^3]^n exp(x*t), where D is the derivative w.r.t. t and D^(-1) t^j/j! = t^(j + 1)/(j + 1)!. E.g., n = 2 generates 20*x*t^3/3! + 90*x^2*t^4/4! + 252*x^3* t^5/5! + ... . For the general unsigned r-Lah number array, replace the threes by (2*r - 1) in the operator. The e.g.f. of the row polynomials is then exp[D^(-1) tDt t^(-(2*r-1))D t^(2*r - 1)] exp(x*t), with offset n = 0. - Tom Copeland, Sep 21 2008

A001551 a(n) = 1^n + 2^n + 3^n + 4^n.

Original entry on oeis.org

4, 10, 30, 100, 354, 1300, 4890, 18700, 72354, 282340, 1108650, 4373500, 17312754, 68711380, 273234810, 1088123500, 4338079554, 17309140420, 69107159370, 276040692700, 1102999460754, 4408508961460, 17623571298330, 70462895745100, 281757423024354

Offset: 0

N. J. A. Sloane

From Wolfdieter Lang, Oct 10 2011: (Start)
a(n) = 2*A196836, n >= 0.
a(n)*(-1)^n, n >= 0, gives the z-sequence of the Sheffer triangle A049459 ((signed) 4-restricted Stirling1) which is the inverse Sheffer triangle of A143496 with offset [0,0](4-restricted Stirling2). See the W. Lang link under A006232 for general Sheffer a- and z-sequences. The a-sequence of every (signed) r-restricted Stirling1 number Sheffer triangle is A027641/A027642 (Bernoulli numbers).
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 813.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 364
C. J. Pita Ruiz V., Some Number Arrays Related to Pascal and Lucas Triangles, J. Int. Seq. 16 (2013) #13.5.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (10,-35,50,-24).

Column 4 of array A103438.

A001551:=-2*(5*z-2)*(5*z**2-5*z+1)/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # conjectured by Simon Plouffe in his 1992 dissertation

Table[Total[Range[4]^n], {n, 0, 40}] (* T. D. Noe, Oct 10 2011 *)

[3**n + sigma(4, n) for n in range(23)]  # Zerinvary Lajos, Jun 04 2009

From Wolfdieter Lang, Oct 10 2011: (Start)
E.g.f.: (1-exp(4*x))/(exp(-x)-1) = Sum_{j=1..4} exp(j*x) (trivial).
O.g.f.: 2*(2-5*x)*(1-5*x+5*x^2)/(Product_{j=1..4} (1-j*x)) (via Laplace transformation of the o.g.f., and partial fraction decomposition backwards). See the Maple Program for the o.g.f. conjecture by Simon Plouffe. This has now been proved. (End)

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cp's OEIS Frontend

A193684 Alternating row sums of Sheffer triangle A143496 (4-restricted Stirling2 numbers).

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A008277 Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.

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A143494 Triangle read by rows: 2-Stirling numbers of the second kind.

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A143495 Triangle read by rows: 3-Stirling numbers of the second kind.

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A193685 5-Stirling numbers of the second kind.

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A003468 Number of minimal 3-covers of a labeled n-set.

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