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A008277 Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n. 353
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Also known as Stirling set numbers and written {n, k}.

S2(n,k) counts partitions of an n-set into k nonempty subsets.

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 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) = diff(x*f_n(x),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., binom(Bell(.,x),n) = x^n/n! (cf. Copeland's 2007 formulas), implying binom(xD,n) = binom(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

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 103ff.

B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

G. Boole, Finite Differences, 5th ed. New York, NY: Chelsea, 1970.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 310.

J. H. Conway and R. K. Guy, The Book of Numbers, Springer, p. 92.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.

S.N. Elaydi, An Introduction to Difference Equations, 3rd ed. Springer, 2005.

H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.7.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 244.

A. D. Korshunov, Asymptotic behavior of Stirling numbers of the second kind. (Russian) Metody Diskret. Analiz. No. 39 (1983), 24-41.

E. Kuz'min and A. I. Shirshov: On the number e, pp. 111-119, eq.(6), in: Kvant Selecta: Algebra and Analysis, I, ed. S. Tabachnikov, Am.Math.Soc., 1999, p. 116, eq. (11).

J. Riordan, An Introduction to Combinatorial Analysis, p. 48.

Mark Shattuck, Combinatorial proofs of some Stirling number formulas,  Preprint (ResearchGate), 2014.

J. Stirling, The Differential Method, London, 1749; see p. 7.

LINKS

T. D. Noe, First 100 rows, flattened

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].

V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.

Tewodros Amdeberhan, Valerio de Angelis and Victor H. Moll, Complementary Bell numbers: arithmetical properties and Wilf's conjecture.

Joerg Arndt, Matters Computational (The Fxtbook), pp. 358-360

J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 201l.

J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv:1307.5624 [math.CO], 2013.

Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv preprint arXiv:1105.3044 [math.CO], 2011.

Hacene Belbachir and Mourad Rahmani, Alternating Sums of the Reciprocals of Binomial Coefficients, Journal of Integer Sequences, Vol. 15 (2012), #12.2.8.

Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.

P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.

P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.

W. E. Bleick, and Peter C. C. Wang, Asymptotics of Stirling numbers of the second kind, Proc. Amer. Math. Soc. 42 (1974), 575-580.

W. E. Bleick, and Peter C. C. Wang, Erratum to: "Asymptotics of Stirling numbers of the second kind" (Proc. Amer. Math. Soc. {42} (1974), 575-580), Proc. Amer. Math. Soc. 48 (1975), 518.

B. A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 42.

Pascal Caron, Jean-Gabriel Luque, Ludovic Mignot, Bruno Patrou, State complexity of catenation combined with a boolean operation: a unified approach, arXiv preprint arXiv:1505.03474 [cs.FL], 2015.

C. Cooper and R. E. Kennedy, Patterns, automata and Stirling numbers of the second kind, Mathematics and Computer Education Journal, 26 (1992), 120-124. [From Peter Bala, Oct 03 2008]

T. Copeland's Shadows of Simplicity, A Class of Differential Operators and the Stirling Numbers, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions, Mathemagical Forests, and Addendum to "Mathemagical Forests"

R. M. Dickau, Stirling numbers of the second kind

Tomislav Došlic, Darko Veljan, Logarithmic behavior of some combinatorial sequences, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019)

G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics, arXiv:quant-ph/0401126, 2004.

FindStat - Combinatorial Statistic Finder, The number of blocks in the set partition

Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

M. L. Glasser, A Generalized Apery Series, Journal of Integer Sequences, Vol. 15 (2012), #12.4.3.

M. Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.

M. Griffiths, Close Encounters with Stirling Numbers of the Second Kind, The Mathematics Teacher, Vol. 106, No. 4, November 2012, pp. 313-317.

J. Gubeladze and J. Love, Vertex maps between simplices, cubes, and crosspolytopes, arXiv preprint arXiv:1304.3775 [math.CO], 2013.

L. C. Hsu, Note on an asymptotic expansion of the n-th difference of zero, Ann. Math. Statistics 19 (1948), 273-277.

Yoshinari Inaba, Hyper-Sums of Powers of Integers and the Akiyama-Tanigawa Matrix, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.7.

Charles Knessl and Joseph B. Keller, Stirling number asymptotics from recursion equations using the ray method, Stud. Appl. Math. 84 (1991), no. 1, 43-56.

Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.

D. E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012.

Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11

S.-M. Ma, T. Mansour, M. Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169 [math.CO], 2013.

T. Manneville, V. Pilaud, Compatibility fans for graphical nested complexes, arXiv preprint arXiv:1501.07152 [math.CO], 2015.

Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.

Nelma Moreira and Rogerio Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]

A. O. Munagi, k- Complementing Subsets of Nonnegative Integers, International Journal of Mathematics and Mathematical Sciences, 2005:2 (2005), 215-224.

OEIS Wiki, Sorting numbers

K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem, arXiv:quant-ph/0312202, 2003.

K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. vol. 50, 083512 (2009)

Feng Qi, An Explicit Formula for Bell Numbers in Terms of Stirling Numbers and Hypergeometric Functions, arXiv:1402.2361 [math.CO], 2014.

S. Ramanujan, Notebook entry

Raymond Scurr and Gloria Olive, Stirling numbers revisited, Discrete Math. 189 (1998), no. 1-3, 209--219. MR1637761 (99d:11019).

Mark Shattuck, Combinatorial proofs of some Stirling number formulas, Pure Mathematics and Applications, Volume 25, Issue 1 (Sep 2015).

A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K. A. Penson, Partition functions and graphs: A combinatorial approach, arXiv:quant-ph/0409082, 2004.

N. M. Temme, Asymptotic estimates of Stirling numbers, Stud. Appl. Math. 89 (1993), no. 3, 233-243.

A. N. Timashev, On asymptotic expansions of Stirling numbers of the first and second kinds. (Russian) Diskret. Mat. 10 (1998), no. 3,148-159 translation in Discrete Math. Appl. 8 (1998), no. 5, 533-544.

A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]

Dennis Walsh, Counting forests with Stirling and Bell numbers

Eric Weisstein's World of Mathematics, Differential Operator and Stirling Number of the Second Kind

Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).

H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, pp. 17ff, 105ff.

M. C. Wolf, Symmetric functions for non-commutative elements, Duke Math. J., 2 (1936), 626-637.

FORMULA

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.

A019538(n, k) = k! * S2(n, k).

A028248(n, k) = (k-1)! * S2(n, k).

For asymptotics see Hsu (1948), among other sources.

The k-th row (k>=1) contains a(n, k) for n=1 to k where a(n, k) = (1/(n-1)!) * Sum_{q=1..[2*n+1+(-1)^(n-1)]/4} [ binomial(n-1, 2*q-2)*(n-2*q+2)^(k-1) - binomial(n-1, 2*q-1)*(n-2*q+1)^(k-1) ]. E.g. Row 7 contains S2(7, 3)=301 { A001298, S2(n+4, n) } and will be computed as the following: S2(7, 3) = a(3, 7) = 1/(3-1)! * Sum_{q=1..2} [ binomial(3-1, 2*q-2)*(3-2*q+2)^(7-1) - binomial(3-1, 2*q-1)*(3-2*q+1)^(7-1) ] = Sum_{q=1..2} [ binomial(2, 2*q-2)*(5-2*q)^6 - binomial(2, 2*q-1)*(4-2*q)^6 ]/2! = [ binomial(2, 0)*3^6 - binomial(2, 1)*2^6 + binomial(2, 2)*1^6 - binomial(2, 3)*0^6 ]/2! = [ 729 - 128 + 1 - 0 ]/2 = 301. - André F. Labossière, Jun 07 2004

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)(Initial index fix, Apr 17 2014)

Bell(n,x) = Sum_{j=1..n} S2(n,j) * x^j = Sum_{j=1..n} E(n,j) * Lag(n,-x, j-n) = Sum_{j=1..n} [ E(n,j)/n! ] * [ n!*Lag(n,-x, j-n) ] = Sum_{j=1..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; Lag(n,x,m), the associated Laguerre polynomials of order m.

By substituting the umbral compositional inverse of the Bell polynomials, the lower factorial n!*binomial(x,n), for x in the equation, the equation becomes x^n = sum(j=1..n) S2(n,j) * j! * binomial(x,j).

Note that E(n,j) / n! = E(n,j) / Sum_{k=1..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=1..n} S2(n,j) = Sum_{j=1..n} E(n,j) * n!*Lag(n,-1, j-n).

(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(x) = 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)

EXAMPLE

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

MAPLE

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);

Stirling2rec := proc(n::integer, k::integer) # Calculate the Stirling numbers of the second kind S2(n, k) by a recurrence. local i, Result; if k > n or n < 0 or k < 0 then return fail end if; if n = 0 or k = 1 then Result := 1; return Result end if; if k = 0 then Result := 0; return Result end if; Result := 0; for i from k to n do Result := Result + binomial(n - 1, i - 1) * Stirling2rec(i - 1, k - 1); end do; return Result; end proc; # Thomas Wieder, Jan 27 2007

MATHEMATICA

Table[StirlingS2[n, k], {n, 11}, {k, n}] // Flatten (* Robert G. Wilson v, May 23 2006 *)

p[x_, n_] = Sum[m^n*x^m/m!, {m, 0, Infinity}]/(x*Exp[x]);

Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];

Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];

Flatten[%] (* Roger L. Bagula, Jan 11 2009 *)

PROG

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

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

(Haskell)

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

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

(Sage) stirling_number2(n, k) # Danny Rorabaugh, Oct 11 2015

(J) n ((] (1 % !)) * +/@((^~ * (] (_1 ^ |.)) * (! {:)@]) i.@>:)) k NB. Stephen Makdisi, Apr 06 2016

CROSSREFS

Cf. A008275 (Stirling numbers of first kind), A048993 (another version of this triangle).

Cf. A000217, A001296, A001297, A001298, A028246, A039810-A039813, A048994, A087107-A087111, A087127, A094262, A127701.

Sequence in context: A250119 A154959 A080417 * A218577 A193387 A185982

Adjacent sequences:  A008274 A008275 A008276 * A008278 A008279 A008280

KEYWORD

nonn,easy,tabl,nice,core,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified April 30 09:01 EDT 2016. Contains 272221 sequences.