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A008275 Triangle read by rows of Stirling numbers of first kind, s(n,k), n >= 1, 1 <= k <= n. 196

%I

%S 1,-1,1,2,-3,1,-6,11,-6,1,24,-50,35,-10,1,-120,274,-225,85,-15,1,720,

%T -1764,1624,-735,175,-21,1,-5040,13068,-13132,6769,-1960,322,-28,1,

%U 40320,-109584,118124,-67284,22449,-4536,546,-36,1,-362880,1026576

%N Triangle read by rows of Stirling numbers of first kind, s(n,k), n >= 1, 1 <= k <= n.

%C The unsigned numbers are also called Stirling cycle numbers: |s(n,k)| = number of permutations of n objects with exactly k cycles.

%C The unsigned numbers (read from right to left) also give the number of permutations of 1..n with complexity k, where the complexity of a permutation is defined to be the sum of the lengths of the cycles minus the number of cycles. In other words, the complexity equals the sum of (length of cycle)-1 over all cycles. For n=5, the numbers of permutations with complexity 0,1,...,4 are 1, 10, 35, 50, 24. - _N. J. A. Sloane_, Feb 08 2019

%C The unsigned numbers are also the number of permutations of 1..n with k left to right maxima (see Khovanova and Lewis, Smith).

%C With P(n) = the number of integer partitions of n, T(i,n) = the number of parts of the i-th partition of n, D(i,n) = the number of different parts of the i-th partition of n, p(j,i,n) = the j-th part of the i-th partition of n, m(j,i,n) = multiplicity of the j-th part of the i-th partition of n, Sum_[T(i,n)=k]_{i=1}^{P(n)} = sum running from i=1 to i=p(n) but taking only partitions with T(i,n)=k parts into account, Product_{j=1..T(i,n)} = product running from j=1 to j=T(i,n), Product_{j=1..D(i,n)} = product running from j=1 to j=D(i,n) one has S1(n,k) = Sum_[T(i,n)=k]_{i=1}^{P(n)} (n!/Product_{j=1..T(i,n)} p(j,i,n))* (1/Product_{j=1..D(i,n)} m(j,i,n)!). For example, S1(6,3) = 225 because n=6 has the following partitions with k=3 parts: (114), (123), (222). Their complexions are: (114): (6!/1*1*4)*(1/2!*1!) = 90, (123): (6!/1*2*3)*(1/1!*1!*1!) = 120, (222): (6!/2*2*2)*(1/3!) = 15. The sum of the complexions is 90+120+15 = 225 = S1(6,3). - _Thomas Wieder_, Aug 04 2005

%C Row sums equal 0. - _Jon Perry_, Nov 14 2005

%C |s(n,k)| enumerates unordered n-vertex forests composed of k increasing non-plane (unordered) trees. Proof from the e.g.f. of the first column and the F. Bergeron et al. reference, especially Table 1, last row (non-plane "recursive"), given in A049029. - _Wolfdieter Lang_, Oct 12 2007

%C |s(n,k)| enumerates unordered increasing n-vertex k-forests composed of k unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j >= 0 come in j+1 colors (j=0 for the k roots). - _Wolfdieter Lang_, Oct 12 2007, Feb 22 2008

%C T(n,k) = A048993(n,k), for k=1..n. - _Reinhard Zumkeller_, Mar 18 2013

%C A refinement of the unsigned array is A036039. For an association to forests of "naturally grown" rooted non-planar trees, dispositions of flags on flagpoles, and colorings of the vertices of the complete graphs K_n, see A130534. - _Tom Copeland_, Mar 30 and Apr 05 2014

%C The Stirling numbers of the first kind were related to the falling factorial and the convolved, or generalized, Bernoulli numbers B_n by Norlund in 1924 through sum[k=1,n+1; T(n+1,k) x^(k-1)] = (x-1)!/(x-1-n)! = (x+B.(0))^n = B_n(x), umbrally evaluated with (B.(0))^k = B_k(0) and the associated Appell polynomial B_n(x) defined by the e.g.f. [t/(e^t-1)]^(n+1) * e^(xt) = exp(B.(x)t). - _Tom Copeland_, Sep 29 2015

%C With x = e^z, D_x = d/dx, D_z = d/dz, and p_n(x) the row polynomials of this entry, x^n (D_x)^n = p_n(D_z) = (D_z)! / (D_z - n)! = (xD_x)! / (xD_x - n)!. - _Tom Copeland_, Nov 27 2015

%C From the operator relation z + Psi(1) + sum_{n > 0} (-1)^n (-1/n) binomial(D,n) = z + Psi(1+D) with D = d/dz and Psi the digamma function, Zeta(n+1) = sum_{k > n-1} (1/k) |S(k,n)| / k! for n > 0 and Zeta the Riemann zeta function. - _Tom Copeland_, Aug 12 2016

%C Let X_1,...,X_n be i.i.d. random variables with exponential distribution having mean = 1. Let Y = max{X_1,...,X_n}. Then (-1)^n*n!/Sum_{k=1..n+1} a(n+1,k) t^(k-1) is the moment generating function of Y. The expectation of Y is the n-th harmonic number. - _Geoffrey Critzer_, Dec 25 2018

%C In the Ewens sampling theory describing the multivariate probability distribution of the sizes of the allelic classes in a sample of size n under the Infinite Alleles Model, |s(n,k)| gives the coefficient in the formula for the probability that a sample of n alleles has exactly k distinct types. - _Noah A Rosenberg_, Feb 10 2019

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.

%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 93ff.

%D B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 32.

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

%D L. Comtet, Advanced Combinatorics, Reidel, 1974; Chapter V, also p. 310.

%D J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 93.

%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.

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

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

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 245. In the second edition, see Chapter 6, especially p. 259.

%D M. Miyata and J. W. Son, On the complexity of permutations and the metric space of bijections, Tensor, 60 (1998), No. 1, 109-116 (MR1768839).

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

%D R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996.

%D J. Stirling, The Differential Method, London, 1749; see p. 10.

%H T. D. Noe, <a href="/A008275/b008275.txt">Rows 1 to 100 of triangle, flattened.</a>

%H B. S. El-Desouky, N. P. Cakic, T. Mansour. Modified approach to generalized Stirling numbers via differential operators, Appl. Math. Lett. 23 (2010), 115-120. doi:<a href="http://doi.org/10.1016/j.aml.2009.08.018">10.1016/j.aml.2009.08.018</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Nikita Alexeev and Peter Zograf, <a href="https://arxiv.org/abs/1111.3061">Hultman numbers, polygon gluings and matrix integrals</a>, arXiv preprint arXiv:1111.3061 [math.PR], 2011.

%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p. 277.

%H J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, <a href="https://arxiv.org/abs/1307.2010">Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure</a>, arXiv:1307.2010 [math.CO], 2013.

%H E. Barcucci, A. Del Lungo and R. Pinzani, <a href="http://dx.doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.

%H Hacene Belbachir and Mourad Rahmani, <a href="http://www.emis.de/journals/JIS/VOL15/Sury/sury42.html">Alternating Sums of the Reciprocals of Binomial Coefficients</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.2.8.

%H J. L. Cereceda, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Cereceda/cereceda2.html">Generalized Akiyama-Tanigawa Algorithm for Hypersums of Powers of Integers</a>, J. Int. Seq. 16 (2013) #13.3.2.

%H J. L. Cereceda, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Cereceda/cereceda7.html">Iterative Procedure for Hypersums of Powers of Integers</a>, Journal of Integer Sequences, 17 (2014), #14.5.3.

%H Ricky X. F. Chen, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Chen/chen11.html">A Note on the Generating Function for the Stirling Numbers of the First Kind</a>, Journal of Integer Sequences, 18 (2015), #15.3.8.

%H T. Copeland, <a href="http://tcjpn.wordpress.com/2015/08/23/a-class-of-differential-operators-and-the-stirling-numbers/">A Class of Differential Operators and the Stirling Numbers</a>, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>, <a href="http://tcjpn.wordpress.com/2011/04/11/lagrange-a-la-lah/">Lagrange a la Lah</a>

%H H. Crane, <a href="https://projecteuclid.org/euclid.ss/1455115906">The ubiquitous Ewens sampling formula</a>, Statistical Science, 31 (2016), 1-19.

%H Thierry Dana-Picard and David G. Zeitoun, <a href="http://www.tandfonline.com/doi/abs/10.1080/0020739X.2011.582172">Sequences of definite integrals, infinite series and Stirling numbers</a>, International Journal of Mathematical Education in Science and Technology, Jun 19 2011.

%H Sajal K. Das, Joydeep Ghosh, and Narsingh Deo, <a href="http://dx.doi.org/10.1016/0166-218X(92)90128-W">Stirling networks: a versatile combinatorial topology for multiprocessor systems</a>, Discrete applied mathematics 37 (1992): 119-146.

%H R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/stirling1.html">Stirling numbers of the first kind</a>

%H Askar Dzhumadil’daev and Damir Yeliussizov, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p10">Walks, partitions, and normal ordering</a>, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000007">The number of saliances of a permutation</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000031">The number of cycles in the cycle decomposition of a permutation</a>.

%H Bill Gosper, <a href="/A008275/a008275.png">Colored illustrations of triangle of Stirling numbers of first kind read mod 2, 3, 4, 5, 6, 7</a>

%H D. B. Gruenberg, <a href="https://arxiv.org/abs/math/0607514">On asymptotics, Stirling numbers, Gamma function and polylogs</a>, arXiv:math/0607514 [math.CO], 2006.

%H J. Hines, <a href="http://www.jstor.org/stable/3029630">A generalization of the S-Stirling numbers</a>, Math. Mag., 29 (1956), 200-203.

%H Yoshinari Inaba, <a href="http://www.emis.de/journals/JIS/VOL8/Inaba/inaba301.html">Hyper-Sums of Powers of Integers and the Akiyama-Tanigawa Matrix</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.7.

%H Charles Knessl and Joseph B. Keller, <a href="http://dx.doi.org/10.1002/sapm199184143">Stirling number asymptotics from recursion equations using the ray method</a>, Stud. Appl. Math. 84 (1991), no. 1, 43-56.

%H Tanya Khovanova and Joel Brewster Lewis, <a href="http://blog.tanyakhovanova.com/?p=451">Skyscrapers</a>

%H D. E. Knuth, <a href="https://arxiv.org/abs/math/9207221">Convolution polynomials</a>, arXiv:math/9207221 [math.CA], 1993; The Mathematica J., 2 (1992), 67-78.

%H W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

%H D. E. Loeb, <a href="https://arxiv.org/abs/math/9502217">A generalization of Stirling numbers</a>, arXiv:math/9502217 [math.CO], 1995.

%H M. M. Mangontarum, J. Katriel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Mangontarum/mango2.html">On q-Boson Operators and q-Analogues of the r-Whitney and r-Dowling Numbers</a>, J. Int. Seq. 18 (2015) 15.9.8.

%H T. Mansour, A. Munagi, M. Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Mansour/mansour4.html">Recurrence Relations and Two-Dimensional Set Partitions</a>, J. Int. Seq. 14 (2011) # 11.4.1

%H Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Schork/schork2.html">The Generalized Stirling and Bell Numbers Revisited</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.

%H B. H. Margolius, <a href="http://dx.doi.org/10.1007/s11134-007-9027-8">Transient and periodic solution to the time-inhomogeneous quasi-birth death process</a>, Queueing Systems, Volume 56, Numbers 3-4 / August, 2007. [From _N. J. A. Sloane_, Jul 09 2009]

%H N. Norlund, <a href="http://tocs.ulb.tu-darmstadt.de/60059613.pdf">Vorlesungen uber Differenzenrechnung</a> Chelsea Pub. Co., New York, 1954.

%H K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, <a href="https://arxiv.org/abs/0904.0369">Laguerre-type derivatives: Dobinski relations and combinatorial identities</a>, J. Math. Phys. vol. 50, 083512 (2009).

%H G. Rzadkowski, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Rzadkowski/rzadkowski3.html">Two formulas for Successive Derivatives and Their Applications</a>, JIS 12 (2009) 09.8.2

%H M. D. Schmidt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html">Generalized j-Factorial Functions, Polynomials, and Applications </a>, J. Int. Seq. 13 (2010), 10.6.7.

%H Raymond Scurr, Gloria Olive, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00011-9">Stirling numbers revisited</a>, Discrete Math. 189 (1998), no. 1-3, 209--219. MR1637761 (99d:11019).

%H Mark Shattuck, <a href="http://www.emis.de/journals/INTEGERS/papers/m59/m59.Abstract.html">Convolution identities for Stirling numbers of the first kind via involution</a>, INTEGERS, 12, 2012, #A59. - From _N. J. A. Sloane_, Feb 04 2013

%H Warren D. Smith, <a href="http://rangevoting.org/PuzzCountRecords.html">Puzzle: Counting new records</a>

%H M. Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Spivey/spivey25.html">A generalized recurrence for Bell Numbers</a>, JIS 11 (2008) 08.2.5

%H M. Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Spivey/spivey31.html">On Solutions to a General Combinatorial Recurrence</a>, J. Int. Seq. 14 (2011) # 11.9.7.

%H R. P. Stanley, <a href="https://arxiv.org/abs/math/0501256">Ordering events in Minkowski space</a>, arXiv:math/0501256 [math.CO], 2005.

%H N. M. Temme, <a href="http://oai.cwi.nl/oai/asset/2304/2304A.pdf">http://algo.inria.fr/seminars/sem92-93/temme.pdf</a>, Stud. Appl. Math. 89 (1993), no. 3, 233-243.

%H A. N., Timashev, <a href="http://dx.doi.org/10.4213/dm440">On asymptotic expansions of Stirling numbers of the first and second kinds</a>, (Russian) Diskret. Mat. 10 (1998), no. 3,148-159 translation in Discrete Math. Appl. 8 (1998), no. 5, 533-544.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PermutationCycle.html">Permutation Cycle</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingNumberoftheFirstKind.html">Stirling Number of the First Kind</a>

%H Thomas Wieder, <a href="/A008275/a008275.txt">Comments on A008275</a>

%H OEIS Wiki, <a href="/wiki/Factorial_polynomials">Factorial polynomials</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F s(n, k)=s(n-1, k-1)-(n-1)*s(n-1, k), n, k >= 1; s(n, 0)=s(0, k)=0; s(0, 0)=1.

%F The unsigned numbers a(n, k)=|s(n, k)| satisfy a(n, k)=a(n-1, k-1)+(n-1)*a(n-1, k), n, k >= 1; a(n, 0)=a(0, k)=0; a(0, 0)=1.

%F E.g.f.: for m-th column (unsigned): ((-log(1-x))^m)/m!.

%F s(n, k) = T(n-1, k-1), n>1 and k>1, where T(n, k) is the triangle [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, -6, -6, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, ...]and DELTA is Deléham's operator defined in A084938. The unsigned numbers are also |s(n, k)| = T(n-1, k-1), for n>0 and k>0, where T(n, k) = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...].

%F Sum[(-1)^(n-i) StirlingS1[n, i] binomial[i, k], {i,0,n}] == (-1)^(n-k) StirlingS1[n+1, k+1]. - Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007

%F G.f.: S(n) = Product_{j=1..n} (x-j) (i.e., (x-1)(x-2)(x-3) = x^3 - 6x^2 + 11x - 6). - _Jon Perry_, Nov 14 2005

%F a(n,k) = s(k,n) = (-1)^(k-n) * S1(k,n) = ( (-1)^(k-n) ) * ( k!/{(n-1)!*2^(k-n)} ) * [ { 1/(k-n)! }*k^(k-n-1) - { (1/6)*(1/(k-n-2)!) }*k^(k-n-2) + { (1/72)*(1/(k-n-4)!) }*k^(k-n-3) - { (1/6480)*(5/(k-n-6)! -36/(k-n-4)!) }*k^(k-n-4) + { (1/155520)*(5/(k-n-8)!-144/(k-n-6)!) }*k^(k-n-5) - { (1/6531840)*(7/(k-n-10)! -504/(k-n-8)!+2304/(k-n-6)!) }*k^(k-n-6) + { (1/1175731200)*(35/(k-n-12)!-5040/(k-n-10)!+87264/(k-n-8)!) }*k^(k-n-7) - { (1/7054387200)*(5/(k-n-14)!-1260/(k-n-12)!+52704/(k-n-10)!-186624/(k-n-8)!) }*k^(k-n-8) + { (1/338610585600)*(5/(k-n-16)!-2016/(k-n-14)!+164736/(k-n-12)!-2156544/(k-n-10)!) }*k^(k-n-9) - ..... ]. - _André F. Labossière_, Mar 27 2006

%F As lower triangular matrices A008277*A008275 = I, the identity matrix. - _Tom Copeland_, Apr 25 2014

%F a(n,k) = s(n,k) = lim_{y -> 0} Sum_{j=0..k} (-1)^j*binomial(k,j)*((-j*y)!/(-j*y-n)!)*y^(-k)/k! = Sum_{j=0..k} (-1)^(n-j)*binomial(k,j)*((j*y - 1 + n)!/(j*y-1)!)*y^(-k)/k!. - _Tom Copeland_, Aug 28 2015

%F From _Daniel Forgues_ Jan 16 2016: (Start)

%F Let x_(0) := 1 (empty product), and for n >= 1:

%F x_(n) := Product_{k=0..n-1} (x-k), called a factorial term (Boole, 1970) or a factorial polynomial (Elaydi, 2005: p. 60), and also

%F x_(-n) := 1 / [Product_{k=0..n-1} (x+k)].

%F Then, for n >= 1:

%F x_(n) = Sum_{k=1..n} T(n,k) * x^k,

%F 1 / [x_(-n)] = Sum_{k=1..n} |T(n,k)| * x^k,

%F x^n = Sum_{k=1..n} A008277(n,k) * x_(k),

%F where A008277(n,k) are Stirling numbers of the second kind.

%F The row sums (of either signed or absolute values) are

%F Sum_{k=1..n} T(n,k) = 0^(n-1),

%F Sum_{k=1..n} |T(n,k)| = T(n+1,1) = n!. (End)

%F s(n,m) = ((-1)^(n-m)/n)*Sum_{i=0..m-1} C(2*n-m-i, m-i-1)*A008517(n-m+1,n-m-i+1). - _Vladimir Kruchinin_, Feb 14 2018

%e |s(3,2)| = 3 for the three unordered 2-forest with 3 vertices and two increasing (nonplane) trees: ((1),(2,3)), ((2),(1,3)), ((3),(1,2)).

%e Triangle begins:

%e 1

%e -1, 1

%e 2, -3, 1

%e -6, 11, -6, 1

%e 24, -50, 35, -10, 1

%e -120, 274, -225, 85, -15, 1

%e 720, -1764, 1624, -735, 175, -21, 1

%e -5040, 13068, -13132, 6769, -1960, 322, -28, 1

%e 40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1

%e Another version of the same triangle, from _Joerg Arndt_, Oct 05 2009:

%e s(n,k) := number of permutations of n elements with exactly k cycles ("Stirling cycle numbers")

%e n| total m=1 2 3 4 5 6 7 8 9

%e -+-----------------------------------------------------

%e 1| 1 1

%e 2| 2 1 1

%e 3| 6 2 3 1

%e 4| 24 6 11 6 1

%e 5| 120 24 50 35 10 1

%e 6| 720 120 274 225 85 15 1

%e 7| 5040 720 1764 1624 735 175 21 1

%e 8| 40320 5040 13068 13132 6769 1960 322 28 1

%e 9| 362880 40320 109584 118124 67284 22449 4536 546 36 1

%e |s(4,2)| = 11 for the eleven unordered 2-forest with 4 vertices, composed of two increasing (nonplane) trees: ((1),((23)(24))), ((2),((13)(14)), ((3),((12)(14)), ((4),((12)(13)); ((1),(2,3,4)),((2),(1,2,3)), ((3), (1,2,4)), ((4),(1,2,3)); ((1,2),(3,4)), ((1,3),(2,4)), ((1,4),(2, 3)). - _Wolfdieter Lang_, Feb 22 2008

%p with (combinat):seq(seq(stirling1(n, k), k=1..n), n=1..10); # _Zerinvary Lajos_, Jun 03 2007

%p for i from 0 to 9 do seq(stirling1(i, j), j = 1 .. i) od; # _Zerinvary Lajos_, Nov 29 2007

%t Flatten[Table[StirlingS1[n, k], {n, 1, 10}, {k, 1, n}]][[1 ;; 47]] (* _Jean-François Alcover_, May 18 2011 *)

%o (PARI) T(n,k)=if(n<1,0,n!*polcoeff(binomial(x,n),k))

%o (PARI) T(n,k)=if(n<1,0,n!*polcoeff(polcoeff((1+x+x*O(x^n))^y,n),k))

%o (PARI) vecstirling(n)=Vec(factorback(vector(n-1,i,1-i*'x))) /* (A function that returns all the s(n,k) as a vector) */ \\ Bill Allombert (Bill.Allombert(AT)math.u-bordeaux1.fr), Mar 16 2009

%o (Maxima) create_list(stirling1(n+1,k+1),n,0,30,k,0,n); /* _Emanuele Munarini_, Jun 01 2012 */

%o (Haskell)

%o a008275 n k = a008275_tabl !! (n-1) !! (k-1)

%o a008275_row n = a008275_tabl !! (n-1)

%o a008275_tabl = map tail $ tail a048994_tabl

%o -- _Reinhard Zumkeller_, Mar 18 2013

%Y Diagonals: A000217, A000914, A001303, A000915, A053567, etc.

%Y Cf. A048994, A008277 (Stirling numbers of second kind), A039814, A039815, A039816, A039817, A048993, A087748.

%Y Cf. A084938, A094216, A008276 (row reversed), A008277, A008278, A094262, A121632, A130534 (unsigned version), A087755 (triangle mod 2), A000142 (row sums of absolute values).

%K sign,tabl,nice,core,changed

%O 1,4

%A _N. J. A. Sloane_

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Last modified February 20 20:55 EST 2019. Contains 320345 sequences. (Running on oeis4.)