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%I #164 Apr 02 2024 11:44:20
%S 1,0,1,0,1,1,0,2,3,1,0,6,11,6,1,0,24,50,35,10,1,0,120,274,225,85,15,1,
%T 0,720,1764,1624,735,175,21,1,0,5040,13068,13132,6769,1960,322,28,1,0,
%U 40320,109584,118124,67284,22449,4536,546,36,1,0,362880,1026576,1172700
%N Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
%C Another name: Triangle of signless Stirling numbers of the first kind.
%C Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,1,2,2,3,3,4,4,5,5,...] DELTA [1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938.
%C A094645*A007318 as infinite lower triangular matrices.
%C Row sums are the factorial numbers. - _Roger L. Bagula_, Apr 18 2008
%C Exponential Riordan array [1/(1-x), log(1/(1-x))]. - _Ralf Stephan_, Feb 07 2014
%C Also the Bell transform of the factorial numbers (A000142). For the definition of the Bell transform see A264428 and for cross-references A265606. - _Peter Luschny_, Dec 31 2015
%C This is the lower triagonal Sheffer matrix of the associated or Jabotinsky type |S1| = (1, -log(1-x)) (see the W. Lang link under A006232 for the notation and references). This implies the e.g.f.s given below. |S1| is the transition matrix from the monomial basis {x^n} to the rising factorial basis {risefac(x,n)}, n >= 0. - _Wolfdieter Lang_, Feb 21 2017
%C T(n, k), for n >= k >= 1, is also the total volume of the n-k dimensional cell (polytope) built from the n-k orthogonal vectors of pairwise different lengths chosen from the set {1, 2, ..., n-1}. See the elementary symmetric function formula for T(n, k) and an example below. - _Wolfdieter Lang_, May 28 2017
%C From _Wolfdieter Lang_, Jul 20 2017: (Start)
%C The compositional inverse w.r.t. x of y = y(t;x) = x*(1 - t(-log(1-x)/x)) = x + t*log(1-x) is x = x(t;y) = ED(y,t) := Sum_{d>=0} D(d,t)*y^(d+1)/(d+1)!, the e.g.f. of the o.g.f.s D(d,t) = Sum_{m>=0} T(d+m, m)*t^m of the diagonal sequences of the present triangle. See the P. Bala link for a proof (there d = n-1, n >= 1, is the label for the diagonals).
%C This inversion gives D(d,t) = P(d, t)/(1-t)^(2*d+1), with the numerator polynomials P(d, t) = Sum_{m=0..d} A288874(d, m)*t^m. See an example below. See also the P. Bala formula in A112007. (End)
%C For n > 0, T(n,k) is the number of permutations of the integers from 1 to n which have k visible digits when viewed from a specific end, in the sense that a higher value hides a lower one in a subsequent position. - _Ian Duff_, Jul 12 2019
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 31, 187, 441, 996.
%D R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., Table 259, p. 259.
%D Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 149-150
%H Reinhard Zumkeller, <a href="/A132393/b132393.txt">Rows n = 0..125 of triangle, flattened</a>
%H Roland Bacher and P. De La Harpe, <a href="https://hal.archives-ouvertes.fr/hal-01285685/document">Conjugacy growth series of some infinitely generated groups</a>, hal-01285685v2, 2016.
%H Eli Bagno and David Garber, <a href="https://arxiv.org/abs/2401.08365">Combinatorics of q,r-analogues of Stirling numbers of type B</a>, arXiv:2401.08365 [math.CO], 2024. See page 5.
%H Peter Bala, <a href="/A112007/a112007_Bala.txt">Diagonals of triangles with generating function exp(t*F(x))</a>.
%H J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, <a href="http://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 Jean-Luc Baril and Sergey Kirgizov, <a href="http://jl.baril.u-bourgogne.fr/Stirling.pdf">The pure descent statistic on permutations</a>, Preprint, 2016.
%H Jean-Luc Baril and Sergey Kirgizov, <a href="https://arxiv.org/abs/2101.01928">Transformation à la Foata for special kinds of descents and excedances</a>, arXiv:2101.01928 [math.CO], 2021.
%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 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 W. S. Gray and M. Thitsa, <a href="http://dx.doi.org/10.1109/SSST.2013.6524939">System Interconnections and Combinatorial Integer Sequences</a>, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
%H John M. Holte, <a href="http://www.jstor.org/stable/2974981">Carries, Combinatorics and an Amazing Matrix</a>, The American Mathematical Monthly, Vol. 104, No. 2 (Feb., 1997), pp. 138-149.
%H Tanya Khovanova and J. B. Lewis, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Khovanova/khova6.html">Skyscraper Numbers</a>, J. Int. Seq. 16 (2013) #13.7.2.
%H Sergey Kitaev and Philip B. Zhang, <a href="https://arxiv.org/abs/1811.07679">Distributions of mesh patterns of short lengths</a>, arXiv:1811.07679 [math.CO], 2018.
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers</a>, arXiv:1707.04451 [math.NT], 2017.
%H Shi-Mei Ma, <a href="http://arxiv.org/abs/1208.3104">Some combinatorial sequences associated with context-free grammars</a>, arXiv:1208.3104v2 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
%H Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.html">Shifting Property for Riordan, Sheffer and Connection Constants Matrices</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
%H Emanuele Munarini, <a href="https://doi.org/10.2298/AADM180226017M">Combinatorial identities involving the central coefficients of a Sheffer matrix</a>, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
%H X.-T. Su, D.-Y. Yang, and W.-W. Zhang, <a href="http://ajc.maths.uq.edu.au/pdf/56/ajc_v56_p133.pdf">A note on the generalized factorial</a>, Australasian Journal of Combinatorics, Volume 56 (2013), Pages 133-137.
%F T(n,k) = T(n-1,k-1)+(n-1)*T(n-1,k), n,k>=1; T(n,0)=T(0,k); T(0,0)=1.
%F Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000142(n), A001147(n), A007559(n), A007696(n), A008548(n), A008542(n), A045754(n), A045755(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - _Philippe Deléham_, Nov 13 2007
%F Expand 1/(1-t)^x = Sum_{n>=0}p(x,n)*t^n/n!; then the coefficients of the p(x,n) produce the triangle. - _Roger L. Bagula_, Apr 18 2008
%F Sum_{k=0..n} T(n,k)*2^k*x^(n-k) = A000142(n+1), A000165(n), A008544(n), A001813(n), A047055(n), A047657(n), A084947(n), A084948(n), A084949(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - _Philippe Deléham_, Sep 18 2008
%F a(n) = Sum_{k=0..n} T(n,k)*3^k*x^(n-k) = A001710(n+2), A001147(n+1), A032031(n), A008545(n), A047056(n), A011781(n), A144739(n), A144756(n), A144758(n) for x=1,2,3,4,5,6,7,8,9,respectively. - _Philippe Deléham_, Sep 20 2008
%F Sum_{k=0..n} T(n,k)*4^k*x^(n-k) = A001715(n+3), A002866(n+1), A007559(n+1), A047053(n), A008546(n), A049308(n), A144827(n), A144828(n), A144829(n) for x=1,2,3,4,5,6,7,8,9 respectively. - _Philippe Deléham_, Sep 21 2008
%F Sum_{k=0..n} x^k*T(n,k) = x*(1+x)*(2+x)*...*(n-1+x), n>=1. - _Philippe Deléham_, Oct 17 2008
%F From _Wolfdieter Lang_, Feb 21 2017: (Start)
%F E.g.f. k-th column: (-log(1 - x))^k, k >= 0.
%F E.g.f. triangle (see the Apr 18 2008 Baluga comment): exp(-x*log(1-z)).
%F E.g.f. a-sequence: x/(1 - exp(-x)). See A164555/A027642. The e.g.f. for the z-sequence is 0. (End)
%F From _Wolfdieter Lang_, May 28 2017: (Start)
%F The row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k, for n >= 0, are R(n, x) = risefac(x,n-1) := Product_{j=0..n-1} x+j, with the empty product for n=0 put to 1. See the Feb 21 2017 comment above. This implies:
%F T(n, k) = sigma^{(n-1)}_(n-k), for n >= k >= 1, with the elementary symmetric functions sigma^{(n-1))_m of degree m in the n-1 symbols 1, 2, ..., n-1, with binomial(n-1, m) terms. See an example below.(End)
%F Boas-Buck type recurrence for column sequence k: T(n, k) = (n!*k/(n - k)) * Sum_{p=k..n-1} beta(n-1-p)*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, and beta(k) = A002208(k+1)/A002209(k+1). See a comment and references in A286718. - _Wolfdieter Lang_, Aug 11 2017
%F T(n,k) = Sum_{j=k..n} j^(j-k)*binomial(j-1, k-1)*A354795(n,j) for n > 0. - _Mélika Tebni_, Mar 02 2023
%F n-th row polynomial: n!*Sum_{k = 0..2*n} (-1)^k*binomial(-x, k)*binomial(-x, 2*n-k) = n!*Sum_{k = 0..2*n} (-1)^k*binomial(1-x, k)*binomial(-x, 2*n-k). - _Peter Bala_, Mar 31 2024
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 2, 3, 1;
%e 0, 6, 11, 6, 1;
%e 0, 24, 50, 35, 10, 1;
%e 0, 120, 274, 225, 85, 15, 1;
%e 0, 720, 1764, 1624, 735, 175, 21, 1;
%e 0, 5040, 13068, 13132, 6769, 1960, 322, 28, 1;
%e ---------------------------------------------------
%e Production matrix is
%e 0, 1
%e 0, 1, 1
%e 0, 1, 2, 1
%e 0, 1, 3, 3, 1
%e 0, 1, 4, 6, 4, 1
%e 0, 1, 5, 10, 10, 5, 1
%e 0, 1, 6, 15, 20, 15, 6, 1
%e 0, 1, 7, 21, 35, 35, 21, 7, 1
%e ...
%e From _Wolfdieter Lang_, May 09 2017: (Start)
%e Three term recurrence: 50 = T(5, 2) = 1*6 + (5-1)*11 = 50.
%e Recurrence from the Sheffer a-sequence [1, 1/2, 1/6, 0, ...]: 50 = T(5, 2) = (5/2)*(binomial(1, 1)*1*6 + binomial(2, 1)*(1/2)*11 + binomial(3, 1)*(1/6)*6 + 0) = 50. The vanishing z-sequence produces the k=0 column from T(0, 0) = 1. (End)
%e Elementary symmetric function T(4, 2) = sigma^{(3)}_2 = 1*2 + 1*3 + 2*3 = 11. Here the cells (polytopes) are 3 rectangles with total area 11. - _Wolfdieter Lang_, May 28 2017
%e O.g.f.s of diagonals: d=2 (third diagonal) [0, 6, 50, ...] has D(2,t) = P(2, t)/(1-t)^5, with P(2, t) = 2 + t, the n = 2 row of A288874. - _Wolfdieter Lang_, Jul 20 2017
%e Boas-Buck recurrence for column k = 2 and n= 5: T(5, 2) = (5!*2/3)*((3/8)*T(2,2)/2! + (5/12)*T(3,2)/3! + (1/2)*T(4,2)/4!) = (5!*2/3)*((3/16 + (5/12)*3/3! + (1/2)*11/4!) = 50. The beta sequence begins: {1/2, 5/12, 3/8, ...}. - _Wolfdieter Lang_, Aug 11 2017
%p a132393_row := proc(n) local k; seq(coeff(expand(pochhammer (x,n)),x,k),k=0..n) end: # _Peter Luschny_, Nov 28 2010
%t p[t_] = 1/(1 - t)^x; Table[ ExpandAll[(n!)SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[(n!)* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a] (* _Roger L. Bagula_, Apr 18 2008 *)
%t Flatten[Table[Abs[StirlingS1[n,i]],{n,0,10},{i,0,n}]] (* _Harvey P. Dale_, Feb 04 2014 *)
%o (Maxima) create_list(abs(stirling1(n,k)),n,0,12,k,0,n); /* _Emanuele Munarini_, Mar 11 2011 */
%o (Haskell)
%o a132393 n k = a132393_tabl !! n !! k
%o a132393_row n = a132393_tabl !! n
%o a132393_tabl = map (map abs) a048994_tabl
%o -- _Reinhard Zumkeller_, Nov 06 2013
%Y Essentially a duplicate of A048994. Cf. A008275, A008277, A112007, A130534, A288874, A354795.
%K nonn,tabl,easy
%O 0,8
%A _Philippe Deléham_, Nov 10 2007, Oct 15 2008, Oct 17 2008
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