A035342 -id:A035342 - cp's OEIS frontend

A008297 Triangle of Lah numbers.

-1, 2, 1, -6, -6, -1, 24, 36, 12, 1, -120, -240, -120, -20, -1, 720, 1800, 1200, 300, 30, 1, -5040, -15120, -12600, -4200, -630, -42, -1, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1, -362880, -1451520, -1693440, -846720, -211680, -28224, -2016, -72, -1, 3628800, 16329600, 21772800, 12700800

Offset: 1

N. J. A. Sloane

|a(n,k)| = number of partitions of {1..n} into k lists, where a list means an ordered subset.
Let N be a Poisson random variable with parameter (mean) lambda, and Y_1,Y_2,... independent exponential(theta) variables, independent of N, so that their density is given by (1/theta)*exp(-x/theta), x > 0. Set S=Sum_{i=1..N} Y_i. Then E(S^n), i.e., the n-th moment of S, is given by (theta^n) * L_n(lambda), n >= 0, where L_n(y) is the Lah polynomial Sum_{k=0..n} |a(n,k)| * y^k. - Shai Covo (green355(AT)netvision.net.il), Feb 09 2010
For y = lambda > 0, formula 2) for the Lah polynomial L_n(y) dated Feb 02 2010 can be restated as follows: L_n(lambda) is the n-th ascending factorial moment of the Poisson distribution with parameter (mean) lambda. - Shai Covo (green355(AT)netvision.net.il), Feb 10 2010
See A111596 for an expression of the row polynomials in terms of an umbral composition of the Bell polynomials and relation to an inverse Mellin transform and a generalized Dobinski formula. - Tom Copeland, Nov 21 2011
Also the Bell transform of the sequence (-1)^(n+1)*(n+1)! without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016
Named after the Slovenian mathematician and actuary Ivo Lah (1896-1979). - Amiram Eldar, Jun 13 2021

			|a(2,1)| = 2: (12), (21); |a(2,2)| = 1: (1)(2). |a(4,1)| = 24: (1234) (24 ways); |a(4,2)| = 36: (123)(4) (6*4 ways), (12)(34) (3*4 ways); |a(4,3)| = 12: (12)(3)(4) (6*2 ways); |a(4,4)| = 1: (1)(2)(3)(4) (1 way).
Triangle:
    -1;
     2,    1;
    -6,   -6,   -1;
    24,   36,   12,   1;
  -120, -240, -120, -20, -1; ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
Shai Covo, The moments of a compound Poisson process with exponential or centered normal jumps, J. Probab. Stat. Sci., Vol. 7, No. 1 (2009), pp. 91-100.
Theodore S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176; the sequence called {!}^{n+}. For a link to this paper see A000262.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
S. Gill Williamson, Combinatorics for Computer Science, Computer Science Press, 1985; see p. 176.

T. D. Noe, Rows n=1..100 of triangle, flattened
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], 2013.
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.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
Tom Copeland, Lagrange a la Lah, 2011.
Siad Daboul, Jan Mangaldan, Michael Z. Spivey, and Peter J. Taylor, The Lah Numbers and the n-th Derivative of exp(1/x), Math. Mag., Vol. 86, No. 1 (2013), pp. 39-47.
Askar Dzhumadil'daev and Damir Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014-2015. [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, Vol. 22, No. 4 (2015), #P4.10.
B. S. El-Desouky, Nenad P.Cakić, and Toufik Mansour, Modified approach to generalized Stirling numbers via differential operators, Appl. Math. Lett., Vol. 23, No. 1 (2010), pp. 115-120.
Sen-Peng Eu, Tung-Shan Fu, Yu-Chang Liang, and Tsai-Lien Wong. On xD-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks. arXiv:1701.00600 [math.CO], 2017.
Milan Janjic, Some classes of numbers and derivatives, JIS, Vol. 12 (2009), pp. 09.8.3
Dmitry Karp and Elena Prilepkina, Generalized Stieltjes transforms: basic aspects, arXiv preprint arXiv:1111.4271 [math.CA], 2011.
Dmitry Karp and Elena Prilepkina, Generalized Stieltjes functions and their exact order, Journal of Classical Analysis, Vol. 1, No. 1 (2012), pp. 53-74. - _N. J. A. Sloane_, Dec 25 2012.
Udita N. Katugampola, A new Fractional Derivative and its Mellin Transform, arXiv preprint arXiv:1106.0965 [math.CA], 2011.
Udita N. Katugampola, Mellin Transforms of the Generalized Fractional Integrals and Derivatives, arXiv preprint arXiv:1112.6031 [math.CA], 2011.
Donald E. Knuth, Convolution polynomials, Mathematica J. 2.1 (1992), no. 4, 67-78; arXiv:math/9207221 [math.CA], 1992.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
Toufik Mansour and Matthias Schork, Generalized Bell numbers and algebraic differential equations, Pure Math. Appl.(PU. MA), Vol. 23, No. 2 (2012), pp. 131-142.
Toufik Mansour, Matthias Schork, and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), Article 12.8.3.
Toufik Mansour and Mark Shattuck, A polynomial generalization of some associated sequences related to set partitions, Periodica Mathematica Hungarica, Vol. 75, No. 2 (December 2017), pp. 398-412.
Toufik Mansour, Augustine Munagi, and Mark Shattuck, Set partitions with colored singleton blocks, Discrete Mathematics Letters, 13. 100, (2024). See p. 100.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics, Vol. 13, No. 2 (2019), pp. 495-517.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Jose L. Ramirez and Mark Shattuck, A (p, q)-Analogue of the r-Whitney-Lah Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.5.6.
Mark Shattuck, Identities for Generalized Whitney and Stirling Numbers, Journal of Integer Sequences, Vol. 20 (2017), Article 17.10.4.
Mark Shattuck, Some formulas for the restricted r-Lah numbers, Annales Mathematicae et Informaticae, Vol. 49 (2018), Eszterházy Károly University Institute of Mathematics and Informatics, pp. 123-140.
Mark Shattuck, Combinatorial Proofs of Some Stirling Number Convolution Formulas, J. Int. Seq., Vol. 25 (2022), Article 22.2.2.
Michael Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq., Vol. 14 (2011) Article 11.9.7.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.
Bao-Xuan Zhu, Total positivity from a generalized cycle index polynomial, arXiv:2006.14485 [math.CO], 2020.

Same as A066667 and A105278 except for signs. Other variants: A111596 (differently signed triangle and (0,0)-based), A271703 (unsigned and (0,0)-based), A089231.
A293125 (row sums) and A000262 (row sums of unsigned triangle).
Columns 1-6 (unsigned): A000142, A001286, A001754, A001755, A001777, A001778.
A002868 gives maximal element (in magnitude) in each row.
A248045 (central terms, negated). A130561 is a natural refinement.
Cf. A007318, A048786, A001263, A008275, A008277, A048993, A094638, A130534.
Cf. A008296, A360177.

a008297 n k = a008297_tabl !! (n-1) !! (k-1)
a008297_row n = a008297_tabl !! (n-1)
a008297_tabl = [-1] : f [-1] 2 where
   f xs i = ys : f ys (i + 1) where
     ys = map negate $
          zipWith (+) ([0] ++ xs) (zipWith (*) [i, i + 1 ..] (xs ++ [0]))
-- Reinhard Zumkeller, Sep 30 2014

A008297 := (n,m) -> (-1)^n*n!*binomial(n-1,m-1)/m!;

a[n_, m_] := (-1)^n*n!*Binomial[n-1, m-1]/m!; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Dec 12 2012, after Maple *)
T[n_, n_] := (-1)^n; T[n_, k_]/;0Oliver Seipel, Dec 06 2024 *)

T(n, m) = (-1)^n*n!*binomial(n-1, m-1)/m!
for(n=1,9, for(m=1,n, print1(T(n,m)", "))) \\ Charles R Greathouse IV, Mar 09 2016

use bigint; use ntheory ":all"; my @L; for my $n (1..9) { push @L, map { stirling($n,$,3)*(-1)**$n } 1..$n; } say join(", ",@L); # _Dana Jacobsen, Mar 16 2017

def A008297_triangle(dim): # computes unsigned T(n, k).
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(2+2*k)*M[n-1,k]+((k+1)*(k+2))*M[n-1,k+1]
    return M
A008297_triangle(9) # Peter Luschny, Sep 19 2012

a(n, m) = (-1)^n*n!*A007318(n-1, m-1)/m!, n >= m >= 1.
a(n+1, m) = (n+m)*a(n, m)+a(n, m-1), a(n, 0) := 0; a(n, m) := 0, n < m; a(1, 1)=1.
a(n, m) = ((-1)^(n-m+1))*L(1, n-1, m-1) where L(1, n, m) is the triangle of coefficients of the generalized Laguerre polynomials n!*L(n, a=1, x). These polynomials appear in the radial l=0 eigen-functions for discrete energy levels of the H-atom.
|a(n, m)| = Sum_{k=m..n} |A008275(n, k)|*A008277(k, m), where A008275 = Stirling numbers of first kind, A008277 = Stirling numbers of second kind. - Wolfdieter Lang
If L_n(y) = Sum_{k=0..n} |a(n, k)|*y^k (a Lah polynomial) then the e.g.f. for L_n(y) is exp(x*y/(1-x)). - Vladeta Jovovic, Jan 06 2001
E.g.f. for the k-th column (unsigned): x^k/(1-x)^k/k!. - Vladeta Jovovic, Dec 03 2002
a(n, k) = (n-k+1)!*N(n, k) where N(n, k) is the Narayana triangle A001263. - Philippe Deléham, Jul 20 2003
From Shai Covo (green355(AT)netvision.net.il), Feb 02 2010: (Start)
We have the following expressions for the Lah polynomial L_n(y) = Sum_{k=0..n} |a(n, k)|*y^k -- exact generalizations of results in A000262 for A000262(n) = L_n(1):
1) L_n(y) = y*exp(-y)*n!*M(n+1,2,y), n >= 1, where M (=1F1) is the confluent hypergeometric function of the first kind;
2) L_n(y) = exp(-y)* Sum_{m>=0} y^m*[m]^n/m!, n>=0, where [m]^n = m*(m+1)*...*(m+n-1) is the rising factorial;
3) L_n(y) = (2n-2+y)L_{n-1}(y)-(n-1)(n-2)L_{n-2}(y), n>=2;
4) L_n(y) = y*(n-1)!*Sum_{k=1..n} (L_{n-k}(y) k!)/((n-k)! (k-1)!), n>=1. (End)
The row polynomials are given by D^n(exp(-x*t)) evaluated at x = 0, where D is the operator (1-x)^2*d/dx. Cf. A008277 and A035342. - Peter Bala, Nov 25 2011
n!C(-xD,n) = Lah(n,:xD:) where C(m,n) is the binomial coefficient, xD= x d/dx, (:xD:)^k = x^k D^k, and Lah(n,x) are the row polynomials of this entry. E.g., 2!C(-xD,2)= 2 xD + x^2 D^2. - Tom Copeland, Nov 03 2012
From Tom Copeland, Sep 25 2016: (Start)
The Stirling polynomials of the second kind A048993 (A008277), i.e., the Bell-Touchard-exponential polynomials B_n[x], are umbral compositional inverses of the Stirling polynomials of the first kind signed A008275 (A130534), i.e., the falling factorials, (x)_n = n! binomial(x,n); that is, umbrally B_n[(x).] = x^n = (B.[x])_n.
An operational definition of the Bell polynomials is (xD_x)^n = B_n[:xD:], where, by definition, (:xD_x:)^n = x^n D_x^n, so (B.[:xD_x:])_n = (xD_x)_n = :xD_x:^n = x^n (D_x)^n.
Let y = 1/x, then D_x = -y^2 D_y; xD_x = -yD_y; and P_n(:yD_y:) = (-yD_y)_n = (-1)^n (1/y)^n (y^2 D_y)^n, the row polynomials of this entry in operational form, e.g., P_3(:yD_y:) = (-yD_y)_3 = (-yD_y) (yD_y-1) (yD_y-2) = (-1)^3 (1/y)^3 (y^2 D_y)^3 = -( 6 :yD_y: +  6 :yD_y:^2 + :yD_y:^3 ) = - ( 6 y D_y + 6 y^2 (D_y)^2 + y^3 (D_y)^3).
Therefore, P_n(y) = e^(-y) P_n(:yD_y:) e^y = e^(-y) (-1/y)^n (y^2 D_y)^n e^y = e^(-1/x) x^n (D_x)^n e^(1/x) = P_n(1/x) and P_n(x) =  e^(-1/x) x^n (D_x)^n e^(1/x) = e^(-1/x) (:x D_x:)^n e^(1/x). (Cf. also A094638.) (End)
T(n,k) = Sum_{j=k..n} (-1)^j*A008296(n,j)*A360177(j,k). - Mélika Tebni, Feb 02 2023

A001497 Triangle of coefficients of Bessel polynomials (exponents in decreasing order).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 15, 15, 6, 1, 105, 105, 45, 10, 1, 945, 945, 420, 105, 15, 1, 10395, 10395, 4725, 1260, 210, 21, 1, 135135, 135135, 62370, 17325, 3150, 378, 28, 1, 2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1, 34459425, 34459425, 16216200, 4729725, 945945, 135135, 13860, 990, 45, 1

Offset: 0

N. J. A. Sloane

The (reverse) Bessel polynomials P(n,x):=Sum_{m=0..n} a(n,m)*x^m, the row polynomials, called Theta_n(x) in the Grosswald reference, solve x*(d^2/dx^2)P(n,x) - 2*(x+n)*(d/dx)P(n,x) + 2*n*P(n,x) = 0.
With the related Sheffer associated polynomials defined by Carlitz as
B(0,x) = 1
B(1,x) = x
B(2,x) = x + x^2
B(3,x) = 3 x + 3 x^2 + x^3
B(4,x) = 15 x + 15 x^2 + 6 x^3 + x^4
... (see Mathworld reference), then P(n,x) = 2^n * B(n,x/2) are the Sheffer polynomials described in A119274. - Tom Copeland, Feb 10 2008
Exponential Riordan array [1/sqrt(1-2x), 1-sqrt(1-2x)]. - Paul Barry, Jul 27 2010
From Vladimir Kruchinin, Mar 18 2011: (Start)
For B(n,k){...} the Bell polynomial of the second kind we have
B(n,k){f', f'', f''', ...} = T(n-1,k-1)*(1-2*x)^(k/2-n), where f(x) = 1-sqrt(1-2*x).
The expansions of the first few rows are:
1/sqrt(1-2*x);
1/(1-2*x)^(3/2), 1/(1-2*x);
3/(1-2*x)^(5/2), 3/(1-2*x)^2, 1/(1-2*x)^(3/2);
15/(1-2*x)^(7/2), 15/(1-2*x)^3, 6/(1-2*x)^(5/2), 1/(1-2*x)^2. (End)
Also the Bell transform of A001147 (whithout column 0 which is 1,0,0,...). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016
Antidiagonals of A099174 are rows of this entry. Dividing each diagonal by its first element generates A054142. - Tom Copeland, Oct 04 2016
The row polynomials p_n(x) of A107102 are (-1)^n B_n(1-x), where B_n(x) are the modified Carlitz-Bessel polynomials above, e.g., (-1)^2 B_2(1-x) = (1-x) + (1-x)^2 = 2 - 3 x + x^2 = p_2(x). - Tom Copeland, Oct 10 2016
a(n-1,m-1) counts rooted unordered binary forests with n labeled leaves and m roots. - David desJardins, Feb 23 2019
From Jianing Song, Nov 29 2021: (Start)
The polynomials P_n(x) = Sum_{k=0..n} T(n,k)*x^k satisfy: P_n(x) - (d/dx)P_n(x) = x*P_{n-1}(x) for n >= 1.
{P(n,x)} are related to the Fourier transform of 1/(1+x^2)^(n+1) and x/(1+x^2)^(n+2):
(i) For n >= 0, real number t, we have Integral_{x=-oo..oo} exp(-i*t*x)/(1+x^2)^(n+1) dx = Pi/(2^n*n!) * P_n(|t|) * exp(-|t|);
(ii) For n >= 0, real number t, we have Integral_{x=-oo..oo} x*exp(-i*t*x)/(1+x^2)^(n+2) dx = Pi/(2^(n+1)*(n+1)!) * ((-t)*P_n(-|t|)) * exp(-|t|). (End)
Suppose that f(x) is an n-times differentiable function defined on (a,b) for 0 <= a < b <= +oo, then for n >= 1, the n-th derivative of f(sqrt(x)) on (a^2,b^2) is Sum_{k=1..n} ((-1)^(n-k)*T(n-1,k-1)*f^(k)(sqrt(x))) / (2^n*x^(n-(k/2))), where f^(k) is the k-th derivative of f. - Jianing Song, Nov 30 2023

			Triangle begins
        1,
        1,       1,
        3,       3,      1,
       15,      15,      6,      1,
      105,     105,     45,     10,     1,
      945,     945,    420,    105,    15,    1,
    10395,   10395,   4725,   1260,   210,   21,   1,
   135135,  135135,  62370,  17325,  3150,  378,  28,  1,
  2027025, 2027025, 945945, 270270, 51975, 6930, 630, 36, 1
Production matrix begins
       1,      1,
       2,      2,      1,
       6,      6,      3,     1,
      24,     24,     12,     4,     1,
     120,    120,     60,    20,     5,    1,
     720,    720,    360,   120,    30,    6,   1,
    5040,   5040,   2520,   840,   210,   42,   7,  1,
   40320,  40320,  20160,  6720,  1680,  336,  56,  8, 1,
  362880, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1
This is the exponential Riordan array A094587, or [1/(1-x),x], beheaded.
- _Paul Barry_, Mar 18 2011

J. Riordan, Combinatorial Identities, Wiley, 1968, p. 77.

T. D. Noe, Rows n = 0..50 of triangle, flattened
Peter Bala, Generalized Dobinski formulas
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, chapter 8.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.
O. Frink and H. L. Krall, A new class of orthogonal polynomials, Trans. Amer. Math. Soc. 65,100-115, 1945. [From _Roger L. Bagula_, Feb 15 2009]
E. Grosswald, Bessel Polynomials, Lecture Notes Math. vol. 698 1978 p. 18.
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) #09.8.3.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
B. Leclerc, Powers of staircase Schur functions and symmetric analogues of Bessel polynomials, Discrete Math., 153 (1996), 213-227.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
Toufik Mansour, Matthias Schork and Mark Shattuck, On the Stirling numbers associated with the meromorphic Weyl algebra, Applied Mathematics Letters, Volume 25, Issue 11, November 2012, Pages 1767-1771. - From _N. J. A. Sloane_, Sep 15 2012
W. Mlotkowski and A. Romanowicz, A family of sequences of binomial type, Probability and Mathematical Statistics, Vol. 33, Fasc. 2 (2013), pp. 401-408.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Feng Qi and Bai-Ni Guo, "Some Properties and Generalizations of the Catalan, Fuss, and Fuss-Catalan Numbers", Mathematical Analysis and Applications : Selected Topics (2018), Wiley, Ch. 5, 101-133.
Feng Qi, X.-T. Shi and F.-F. Liu, Several formulas for special values of the Bell polynomials of the second kind and applications, Preprint 2015.
Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233. Lemma 2.2.
Eric Weisstein's World of Mathematics, Bessel Polynomial
Index entries for sequences related to Bessel functions or polynomials

Reflected version of A001498 which is considered the main entry.
Other versions of this same triangle are given in A144299, A111924 and A100861.
Row sums give A001515. a(n, 0)= A001147(n) (double factorials).
Cf. A104556 (matrix inverse). A039683, A122850.
Cf. A245066 (central terms).
Cf. A054142, A099174, A107102.

a001497 n k = a001497_tabl !! n !! k
a001497_row n = a001497_tabl !! n
a001497_tabl = [1] : f [1] 1 where
   f xs z = ys : f ys (z + 2) where
     ys = zipWith (+) ([0] ++ xs) (zipWith (*) [z, z-1 ..] (xs ++ [0]))
-- Reinhard Zumkeller, Jul 11 2014

/* As triangle */ [[Factorial(2*n-k)/(Factorial(k)*Factorial(n-k)*2^(n-k)): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 12 2015

f := proc(n) option remember; if n <=1 then (1+x)^n else expand((2*n-1)*x*f(n-1)+f(n-2)); fi; end;
row := n -> seq(coeff(f(n), x, n - k), k = 0..n): seq(row(n), n = 0..9);

m = 9; Flatten[ Table[(n + k)!/(2^k*k!*(n - k)!), {n, 0, m}, {k, n, 0, -1}]] (* Jean-François Alcover, Sep 20 2011 *)
y[n_, x_] := Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n-1/2, 1/x]; t[n_, k_] := Coefficient[y[n, x], x, k]; Table[t[n, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 01 2013 *)

T(k, n) = if(n>k||k<0||n<0,0,(2*k-n)!/(n!*(k-n)!*2^(k-n))) /* Ralf Stephan */

{T(n, k) = if( k<0 || k>n, 0, binomial(n, k)*(2*n-k)!/2^(n-k)/n!)}; /* Michael Somos, Oct 03 2006 */

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: A001147(n), 9) # Peter Luschny, Jan 19 2016

a(n, m) = (2*n-m)!/(m!*(n-m)!*2^(n-m)) if n >= m >= 0 else 0 (from Grosswald, p. 7).
a(n, m)= 0, n= m >= 0 (from Grosswald p. 23, (19)).
E.g.f. for m-th column: ((1-sqrt(1-2*x))^m)/(m!*sqrt(1-2*x)).
G.f.: 1/(1-xy-x/(1-xy-2x/(1-xy-3x/(1-xy-4x/(1-.... (continued fraction). - Paul Barry, Jan 29 2009
T(n,k) = if(k<=n, C(2n-k,2(n-k))*(2(n-k)-1)!!,0) = if(k<=n, C(2n-k,2(n-k))*A001147(n-k),0). - Paul Barry, Mar 18 2011
Row polynomials for n>=1 are given by 1/t*D^n(exp(x*t)) evaluated at x = 0, where D is the operator 1/(1-x)*d/dx. - Peter Bala, Nov 25 2011
The matrix product A039683*A008277 gives a signed version of this triangle. Dobinski-type formula for the row polynomials: R(n,x) = (-1)^n*exp(x)*Sum_{k = 0..inf} k*(k-2)*(k-4)*...*(k-2*(n-1))*(-x)^k/k!. Cf. A122850. - Peter Bala, Jun 23 2014

A105278 Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 24, 36, 12, 1, 120, 240, 120, 20, 1, 720, 1800, 1200, 300, 30, 1, 5040, 15120, 12600, 4200, 630, 42, 1, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1, 3628800, 16329600

Offset: 1

Miklos Kristof, Apr 25 2005

T(n,k) is the number of partially ordered sets (posets) on n elements that consist entirely of k chains. For example, T(4, 3)=12 since there are exactly 12 posets on {a,b,c,d} that consist entirely of 3 chains. Letting ab denote a<=b and using a slash "/" to separate chains, the 12 posets can be given by a/b/cd, a/b/dc, a/c/bd, a/c/db, a/d/bc, a/d/cb, b/c/ad, b/c/da, b/d/ac, b/d/ca, c/d/ab and c/d/ba, where the listing of the chains is arbitrary (e.g., a/b/cd = a/cd/b =...cd/b/a). - Dennis P. Walsh, Feb 22 2007
Also the matrix product |S1|.S2 of Stirling numbers of both kinds.
This Lah triangle is a lower triangular matrix of the Jabotinsky type. See the column e.g.f. and the D. E. Knuth reference given in A008297. - Wolfdieter Lang, Jun 29 2007
The infinitesimal matrix generator of this matrix is given in A132710. See A111596 for an interpretation in terms of circular binary words and generalized factorials. - Tom Copeland, Nov 22 2007
Three combinatorial interpretations: T(n,k) is (1) the number of ways to split [n] = {1,...,n} into a collection of k nonempty lists ("partitions into sets of lists"), (2) the number of ways to split [n] into an ordered collection of n+1-k nonempty sets that are noncrossing ("partitions into lists of noncrossing sets"), (3) the number of Dyck n-paths with n+1-k peaks labeled 1,2,...,n+1-k in some order. - David Callan, Jul 25 2008
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 = D where D(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. - Tom Copeland, Aug 21 2008
An e.g.f. for the row polynomials of A(n,k) = T(n,k)*a(n-k) is exp[a(.)* D_x * x^2] exp(x*t) = exp(x*t) exp[(.)!*Lag(.,-x*t,1)*a(.)*x], umbrally, where [(.)! Lag(.,x,1)]^n = n! Lag(n,x,1) is a normalized Laguerre polynomial of order 1. - Tom Copeland, Aug 29 2008
Triangle of coefficients from the Bell polynomial of the second kind for f = 1/(1-x). B(n,k){x1,x2,x3,...} = B(n,k){1/(1-x)^2,...,(j-1)!/(1-x)^j,...} = T(n,k)/(1-x)^(n+k). - Vladimir Kruchinin, Mar 04 2011
The triangle, with the row and column offset taken as 0, is the generalized Riordan array (exp(x), x) with respect to the sequence n!*(n+1)! as defined by Wang and Wang (the generalized Riordan array (exp(x), x) with respect to the sequence n! is Pascal's triangle A007318, and with respect to the sequence n!^2 is A021009 unsigned). - Peter Bala, Aug 15 2013
For a relation to loop integrals in QCD, see p. 33 of Gopakumar and Gross and Blaizot and Nowak. - Tom Copeland, Jan 18 2016
Also the Bell transform of (n+1)!. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
Also the number of k-dimensional flats of the n-dimensional Shi arrangement. - Shuhei Tsujie, Apr 26 2019
The numbers T(n,k) appear as coefficients when expanding the rising factorials (x)^k = x(x+1)...(x+k-1) in the basis of falling factorials (x)k = x(x-1)...(x-k+1). Specifically, (x)^n = Sum{k=1..n} T(n,k) (x)k. - _Jeremy L. Martin, Apr 21 2021

			T(1,1) = C(1,1)*0!/0! = 1,
T(2,1) = C(2,1)*1!/0! = 2,
T(2,2) = C(2,2)*1!/1! = 1,
T(3,1) = C(3,1)*2!/0! = 6,
T(3,2) = C(3,2)*2!/1! = 6,
T(3,3) = C(3,3)*2!/2! = 1,
Sheffer a-sequence recurrence: T(6,2)= 1800 = (6/3)*120 + 6*240.
B(n,k) =
   1/(1-x)^2;
   2/(1-x)^3,  1/(1-x)^4;
   6/(1-x)^4,  6/(1-x)^5,  1/(1-x)^6;
  24/(1-x)^5, 36/(1-x)^6, 12/(1-x)^7, 1/(1-x)^8;
The triangle T(n,k) begins:
  n\k      1       2       3      4      5     6    7  8  9 ...
  1:       1
  2:       2       1
  3:       6       6       1
  4:      24      36      12      1
  5:     120     240     120     20      1
  6:     720    1800    1200    300     30     1
  7:    5040   15120   12600   4200    630    42    1
  8:   40320  141120  141120  58800  11760  1176   56  1
  9:  362880 1451520 1693440 846720 211680 28224 2016 72  1
  ...
Row n=10: [3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1]. - _Wolfdieter Lang_, Feb 01 2013
From _Peter Bala_, Feb 24 2025: (Start)
The array factorizes as an infinite product (read from right to left):
  /  1                \        /1             \^m /1           \^m /1           \^m
  |  2    1            |      | 0   1          |  |0  1         |  |1  1         |
  |  6    6   1        | = ...| 0   0   1      |  |0  1  1      |  |0  2  1      |
  | 24   36  12   1    |      | 0   0   1  1   |  |0  0  2  1   |  |0  0  3  1   |
  |120  240 120  20   1|      | 0   0   0  2  1|  |0  0  0  3  1|  |0  0  0  4  1|
  |...                 |      |...             |  |...          |  |...          |
where m = 2. Cf. A008277 (m = 1), A035342 (m = 3), A035469 (m = 4), A049029 (m = 5) A049385 (m = 6), A092082 (m = 7), A132056 (m = 8), A223511 - A223522 (m = 9 through 20), A001497 (m = -1), A004747 (m = -2), A000369 (m = -3), A011801 (m = -4), A013988 (m = -5). (End)

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Peter Bala, Factorising (r,b)-Stirling arrays
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], 2013.
Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, arXiv preprint arXiv:1105.3043 [math.CO], 2011, J. Int. Seq. 14 (2011) # 11.9.5
Jean-Paul Blaizot and Maciej A. Nowak, Large N_c confinement and turbulence, arXiv:0801.1859 [hep-th], 2008.
David Callan, Sets, Lists and Noncrossing Partitions, arXiv:0711.4841 [math.CO], 2007-2008.
Pietro Codara, Ottavio M. D'Antona, and Pavol Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
Tom Copeland, Mathemagical Forests, Addendum to Mathemagical Forests, The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions, A Class of Differential Operators and the Stirling Numbers
Siad Daboul, Jan Mangaldan, Michael Z. Spivey and Peter Taylor, The Lah Numbers and the n-th Derivative of exp(1/x), Math. Mag., 86 (2013), 39-47.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 33.
G. H. E. Duchamp et al., Feynman graphs and related Hopf algebras, J. Phys. (Conf Ser) 30 (2006) 107-118.
Rajesh Gopakumar and David J. Gross, Mastering the master field, arXiv:hep-th/9411021, 1994.
Gábor Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4. - From _Tom Copeland_, Oct 01 2015
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Donald E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
MacTutor History of Mathematics archive: Biography of Ivo Lah.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019. See p. 18.
Michael Penn, Lah Numbers and an appearance of exponential generating functions, YouTube video, 2025.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019. See p. 4.
Tilman Piesk, Illustration of the first four rows
Kornelia Ufniarz and Grzegorz Siudem, Combinatorial origins of the canonical ensemble, arXiv:2008.00244 [math-ph], 2020. See p. 5.
Weiping Wang and Tianming Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
Wikipedia, Lah number

Cf. A000262, A103194, A105220.
Triangle of Lah numbers (A008297) unsigned.
Cf. A111596 (signed triangle with extra n=0 row and m=0 column).
Cf. A130561 (for a natural refinement).
Cf. A094638 (for differential operator representation).
Cf. A248045 (central terms), A002868 (row maxima).
Cf, A059110.
Cf. A001263, A008277.
Cf. A089231 (triangle with mirrored rows).
Cf. A271703 (triangle with extra n=0 row and m=0 column).

Flat(List([1..10],n->List([1..n],k->Binomial(n,k)*Factorial(n-1)/Factorial(k-1)))); # Muniru A Asiru, Jul 25 2018

a105278 n k = a105278_tabl !! (n-1) !! (k-1)
a105278_row n = a105278_tabl !! (n-1)
a105278_tabl = [1] : f [1] 2 where
   f xs i = ys : f ys (i + 1) where
     ys = zipWith (+) ([0] ++ xs) (zipWith (*) [i, i + 1 ..] (xs ++ [0]))
-- Reinhard Zumkeller, Sep 30 2014, Mar 18 2013

/* As triangle */ [[Binomial(n,k)*Factorial(n-1)/Factorial(k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Oct 31 2014

The triangle: for n from 1 to 13 do seq(binomial(n,k)*(n-1)!/(k-1)!,k=1..n) od;
the sequence: seq(seq(binomial(n,k)*(n-1)!/(k-1)!,k=1..n),n=1..13);
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ...) as column 0.
BellMatrix(n -> (n+1)!, 9); # Peter Luschny, Jan 27 2016

nn = 9; a = x/(1 - x); f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Drop[Range[0, nn]! CoefficientList[Series[Exp[y a], {x, 0, nn}], {x, y}], 1]]] (* Geoffrey Critzer, Dec 11 2011 *)
nn = 9; Flatten[Table[(j - k)! Binomial[j, k] Binomial[j - 1, k - 1], {j, nn}, {k, j}]] (* Jan Mangaldan, Mar 15 2013 *)
rows = 10;
t = Range[rows]!;
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
T[n_, n_] := 1; T[n_, k_] /;0Oliver Seipel, Dec 06 2024 *)

use ntheory ":all"; say join ", ", map { my $n=$; map { stirling($n,$,3) } 1..$n; } 1..9; # Dana Jacobsen, Mar 16 2017

T(n,k) = Sum_{m=n..k} |S1(n,m)|*S2(m,k), k>=n>=1, with Stirling triangles S2(n,m):=A048993 and S1(n,m):=A048994.
T(n,k) = C(n,k)*(n-1)!/(k-1)!.
Sum_{k=1..n} T(n,k) = A000262(n).
n*Sum_{k=1..n} T(n,k) = A103194(n) = Sum_{k=1..n} T(n,k)*k^2.
E.g.f. column k: (x^(k-1)/(1-x)^(k+1))/(k-1)!, k>=1.
Recurrence from Sheffer (here Jabotinsky) a-sequence [1,1,0,...] (see the W. Lang link under A006232): T(n,k)=(n/k)*T(n-1,m-1) + n*T(n-1,m). - Wolfdieter Lang, Jun 29 2007
The e.g.f. is, umbrally, exp[(.)!* L(.,-t,1)*x] = exp[t*x/(1-x)]/(1-x)^2 where L(n,t,1) = Sum_{k=0..n} T(n+1,k+1)*(-t)^k = Sum_{k=0..n} binomial(n+1,k+1)* (-t)^k / k! is the associated Laguerre polynomial of order 1. - Tom Copeland, Nov 17 2007
For this Lah triangle, the n-th row polynomial is given umbrally by
  n! C(B.(x)+1+n,n) = (-1)^n C(-B.(x)-2,n), where C(x,n)=x!/(n!(x-n)!),
  the binomial coefficient, and B_n(x)= exp(-x)(xd/dx)^n exp(x), the n-th Bell / Touchard / exponential polynomial (cf. A008277). E.g.,
  2! C(-B.(-x)-2,2) = (-B.(x)-2)(-B.(x)-3) = B_2(x) + 5*B_1(x) + 6 = 6 + 6x + x^2.
  n! C(B.(x)+1+n,n) = n! e^(-x) Sum_{j>=0} C(j+1+n,n)x^j/j! is a corresponding Dobinski relation. See the Copeland link for the relation to inverse Mellin transform. - Tom Copeland, Nov 21 2011
The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^2*d/dx. Cf. A008277 (D = (1+x)*d/dx), A035342 (D = (1+x)^3*d/dx), A035469 (D = (1+x)^4*d/dx) and A049029 (D = (1+x)^5*d/dx). - Peter Bala, Nov 25 2011
T(n,k) = Sum_{i=k..n} A130534(n-1,i-1)*A008277(i,k). - Reinhard Zumkeller, Mar 18 2013
Let E(x) = Sum_{n >= 0} x^n/(n!*(n+1)!). Then a generating function is exp(t)*E(x*t) = 1 + (2 + x)*t + (6 + 6*x + x^2)*t^2/(2!*3!) + (24 + 36*x + 12*x^2 + x^3)*t^3/(3!*4!) + ... . - Peter Bala, Aug 15 2013
P_n(x) = L_n(1+x) = n!*Lag_n(-(1+x);1), where P_n(x) are the row polynomials of A059110; L_n(x), the Lah polynomials of A105278; and Lag_n(x;1), the Laguerre polynomials of order 1. These relations follow from the relation between the iterated operator (x^2 D)^n and ((1+x)^2 D)^n with D = d/dx. - Tom Copeland, Jul 23 2018
Dividing each n-th diagonal by n!, where the main diagonal is n=1, generates the Narayana matrix A001263. - Tom Copeland, Sep 23 2020
T(n,k) = A089231(n,n-k). - Ron L.J. van den Burg, Dec 12 2021
T(n,k) = T(n-1,k-1) + (n+k-1)*T(n-1,k). - Bérénice Delcroix-Oger, Jun 25 2025

Stirling comments and e.g.f.s from Wolfdieter Lang, Apr 11 2007

A049029 Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.

Original entry on oeis.org

1, 5, 1, 45, 15, 1, 585, 255, 30, 1, 9945, 5175, 825, 50, 1, 208845, 123795, 24150, 2025, 75, 1, 5221125, 3427515, 775845, 80850, 4200, 105, 1, 151412625, 108046575, 27478710, 3363045, 219450, 7770, 140, 1, 4996616625, 3824996175, 1069801425

Offset: 1

Wolfdieter Lang

Previous name was: Triangle of numbers related to triangle A048882; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...
a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing quintic (5-ary) trees. Proof based on the a(n,m) recurrence. See also the F. Bergeron et al. reference, especially Table 1, first row and Example 1 for the e.g.f. for m=1. - Wolfdieter Lang, Sep 14 2007
Also the Bell transform of A007696(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle starts:
{1};
{5,1};
{45,15,1};
{585,255,30,1};
{9945,5175,825,50,1};
...

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
T. Copeland, Mathemagical Forests
T. Copeland, Addendum to Mathemagical Forests
T. Copeland, A Class of Differential Operators and the Stirling Numbers
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) #09.8.3.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First 10 rows.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO]. - From N. J. A. Sloane, Aug 21 2012
E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's board, Discr. Maths. 239 (2001) 33-51.
Mathias Pétréolle, Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.

a(n, m) := S2(5, n, m) is the fifth triangle of numbers in the sequence S2(1, n, m) := A008277(n, m) (Stirling 2nd kind), S2(2, n, m) := A008297(n, m) (Lah), S2(3, n, m) := A035342(n, m), S2(4, n, m) := A035469(n, m). a(n, 1)= A007696(n). A007559(n).
Cf. A048882, A007696. Row sums: A049120(n), n >= 1.
Cf. A094638

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> mul(4*k+1, k=0..n), 9); # Peter Luschny, Jan 28 2016

a[n_, m_] /; n >= m >= 1 := a[n, m] = (4(n-1) + m)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* _Jean-François Alcover, Jul 22 2011 *)
rows = 9;
a[n_, m_] := BellY[n, m, Table[Product[4k+1, {k, 0, j}], {j, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

a(n, m) = n!*A048882(n, m)/(m!*4^(n-m)); a(n+1, m) = (4*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n

a(n, m) = sum(|A051142(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. to W. Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.

From Peter Bala, Nov 25 2011: (Start)

E.g.f.: G(x,t) = exp(t*A(x)) = 1+t*x+(5*t+t^2)*x^2/2!+(45*t+15*t^2+t^3)*x^3/3!+..., where A(x) = -1+(1-4*x)^(-1/4) satisfies the autonomous differential equation A'(x) = (1+A(x))^5.

The generating function G(x,t) satisfies the partial differential equation t*(dG/dt+G) = (1-4*x)*dG/dx, from which follows the recurrence given above.

The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^5*d/dx. Cf. A008277 (D = (1+x)*d/dx), A105278 (D = (1+x)^2*d/dx), A035342 (D = (1+x)^3*d/dx) and A035469 (D = (1+x)^4*d/dx).

(End)

New name from Peter Luschny, Jan 30 2016

A035469 Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.

Original entry on oeis.org

1, 4, 1, 28, 12, 1, 280, 160, 24, 1, 3640, 2520, 520, 40, 1, 58240, 46480, 11880, 1280, 60, 1, 1106560, 987840, 295960, 40040, 2660, 84, 1, 24344320, 23826880, 8090880, 1296960, 109200, 4928, 112, 1, 608608000, 643843200

Offset: 1

Wolfdieter Lang

Previous name was: Triangle of numbers related to triangle A035529; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297 and A035342.
a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing quartic (4-ary) trees. Proof based on the a(n,m) recurrence. See a D. Callan comment on the m=1 case A007559. See also the F. Bergeron et al. reference, especially Table 1, first row and Example 1 for the e.g.f. for m=1. - Wolfdieter Lang, Sep 14 2007
For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle starts:
     {1}
     {4,    1}
    {28,   12,    1}
   {280,  160,   24,    1}
  {3640, 2520,  520,   40,    1}

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.

Peter Bala, Generalized Dobinski formulas
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Richell O. Celeste, Roberto B. Corcino and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers
Tom Copeland, Mathemagical Forests
Tom Copeland, Addendum to Mathemagical Forests
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Wolfdieter Lang, First 10 rows.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
E. Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 (2001) 33-51.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.

a(n, m)=: S2(4, n, m) is the fourth triangle of numbers in the sequence S2(1, n, m) := A008277(n, m) (Stirling 2nd kind), S2(2, n, m) := A008297(n, m) (Lah), S2(3, n, m) := A035342(n, m). a(n, 1)= A007559(n).
Row sums: A049119(n), n >= 1.
Cf. A094638.

a[n_, m_] /; n >= m >= 1 := a[n, m] = (3(n-1) + m)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* _Jean-François Alcover, Jul 22 2011 *)
rows = 9;
a[n_, m_] := BellY[n, m, Table[Product[3k+1, {k, 0, j}], {j, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: A007559(n+1) , 9) # Peter Luschny, Jan 19 2016

a(n, m) = Sum_{j=m..n} |A051141(n, j)|*S2(j, m) (matrix product), with S2(j, m):=A008277(j, m) (Stirling2 triangle). Priv. comm. to Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.
a(n, m) = n!*A035529(n, m)/(m!*3^(n-m)); a(n+1, m) = (3*n+m)*a(n, m) + a(n, m-1), n >= m >= 1; a(n, m) := 0, n < m; a(n, 0) := 0, a(1, 1)=1;
E.g.f. of m-th column: ((-1+(1-3*x)^(-1/3))^m)/m!.
From Peter Bala, Nov 25 2011: (Start)
E.g.f.: G(x,t) = exp(t*A(x)) = 1 + t*x + (4*t+t^2)*x^2/2! + (28*t + 12*t^2 + t^3)*x^3/3! + ..., where A(x) = -1 + (1-3*x)^(-1/3) satisfies the autonomous differential equation A'(x) = (1+A(x))^4.
The generating function G(x,t) satisfies the partial differential equation t*(dG/dt+G) = (1-3*x)*dG/dx, from which follows the recurrence given above.
The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^4*d/dx. Cf. A008277 (D = (1+x)*d/dx), A105278 (D = (1+x)^2*d/dx), A035342 (D = (1+x)^3*d/dx) and A049029 (D = (1+x)^5*d/dx).
(End)
Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k>=0} k*(k+3)*(k+6)*...*(k+3*(n-1))*x^k/k!. - Peter Bala, Jun 23 2014

New name from Peter Luschny, Jan 19 2016

A039755 Triangle of B-analogs of Stirling numbers of the second kind.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 13, 9, 1, 1, 40, 58, 16, 1, 1, 121, 330, 170, 25, 1, 1, 364, 1771, 1520, 395, 36, 1, 1, 1093, 9219, 12411, 5075, 791, 49, 1, 1, 3280, 47188, 96096, 58086, 13776, 1428, 64, 1, 1, 9841, 239220, 719860, 618870, 209622, 32340, 2388, 81, 1, 1

Offset: 0

Ruedi Suter (suter(AT)math.ethz.ch)

Let M be an infinite lower triangular bidiagonal matrix with (1,3,5,7,...) in the main diagonal and (1,1,1,...) in the subdiagonal. n-th row = M^n * [1,0,0,0,...]. - Gary W. Adamson, Apr 13 2009
From Peter Bala, Aug 08 2011: (Start)
A type B_n set partition is a partition P of the set {1, 2, ..., n, -1, -2, ..., -n} such that for any block B of P, -B is also a block of P, and there is at most one block, called a zero-block, satisfying B = -B. We call (B, -B) a block pair of P if B is not a zero-block. Then T(n,k) is the number of type B_n set partitions with k block pairs. See [Wang].
For example, T(2,1) = 4 since the B_2 set partitions with 1 block pair are {1,2}{-1,-2}, {1,-2}{-1,2}, {1,-1}{2}{-2} and {2,-2}{1}{-1} (the last two partitions contain a zero block).
(End)
Exponential Riordan array [exp(x), (1/2)*(exp(2*x) - 1)]. Triangle of connection constants for expressing the monomial polynomials x^n as a linear combination of the basis polynomials (x-1)*(x-3)*...*(x-(2*k-1)) of A039757. An example is given below. Inverse array is A039757. Equals matrix product A008277 * A122848. - Peter Bala, Jun 23 2014
T(n, k) also gives the (dimensionless) volume of the multichoose(k+1, n-k) = binomial(n, k) polytopes of dimension n-k with side lengths from the set {1, 3, ..., 1+2*k}. See the column g.f.s and the complete homogeneous symmetric function formula for T(n, k) below. - Wolfdieter Lang, May 26 2017
T(n, k) is the number of k-dimensional subspaces (i.e., sets of fixed points like rotation axes and symmetry planes) of the n-cube. See "Sets of fixed points..." in LINKS section. - Tilman Piesk, Oct 26 2019

			Triangle T(n,k) begins:
  n\k 0     1       2        3       4       5      6     7    8   9 10 ...
  0:  1
  1:  1     1
  2:  1     4       1
  3:  1    13       9        1
  4:  1    40      58       16       1
  5:  1   121     330      170      25       1
  6:  1   364    1771     1520     395      36      1
  7:  1  1093    9219    12411    5075     791     49     1
  8:  1  3280   47188    96096   58086   13776   1428    64    1
  9:  1  9841  239220   719860  618870  209622  32340  2388   81   1
 10:  1 29524 1205941  5278240 6289690 2924712 630042 68160 3765 100  1
 ... reformatted and extended by _Wolfdieter Lang_, May 26 2017
The sequence of row polynomials of A214406 begins [1, 1+x, 1+8*x+3*x^2, ...]. The o.g.f.'s for the diagonals of this triangle thus begin
1/(1-x) = 1 + x + x^2 + x^3 + ...
(1+x)/(1-x)^3 = 1 + 4*x + 9*x^2 + 16*x^3 + ...
(1+8*x+3*x^2)/(1-x)^5 = 1 + 13*x + 58*x^2 + 170*x^3 + ... . - _Peter Bala_, Jul 20 2012
Connection constants: x^3 = 1 + 13*(x-1) + 9*(x-1)*(x-3) + (x-1)*(x-3)*(x-5). Hence row 3 = [1,13,9,1]. - _Peter Bala_, Jun 23 2014
Complete homogeneous symmetric functions: T(3, 1) = h^{(2)}_2 = 1^2 + 3^2 + 1^1*3^1 = 13. The three 2D polytopes are two squares and a rectangle. T(3, 2) = h^{(3)}_1 = 1^1 + 3^1 + 5^1 = 9. The 1D polytopes are three lines. - _Wolfdieter Lang_, May 26 2017
T(4, 3) = 16 is the number of 3-dimensional subspaces (mirror hyperplanes) of the 4-cube. (These are 4 cubes and 12 cuboids.) See "Sets of fixed points..." in LINKS section. - _Tilman Piesk_, Oct 26 2019

G. C. Greubel, Rows = 0..100 of triangle, flattened
V. E. Adler, Set partitions and integrable hierarchies, arXiv:1510.02900 [nlin.SI], 2015.
Alnour Altoum, Hasan Arslan, and Mariam Zaarour, Cauchy numbers in type B, arXiv:2312.14652 [math.CO], 2023.
Hasan Arslan, Nazmiye Alemdar, Mariam Zaarour, and Hüseyin Altındiş, On Bell numbers of type D, arXiv:2504.16522 [math.CO], 2025. See p. 3.
Eli Bagno, Riccardo Biagioli and David Garber, Some identities involving second kind Stirling numbers of types B and D, arXiv:1901.07830 [math.CO], 2019.
Peter Bala, Generalized Dobinski formulas
Paul Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2.
Sandrine Dasse-Hartaut and Pawel Hitczenko, Greek letters in random staircase tableaux arXiv:1202.3092v1 [math.CO], 2012.
I. Dolgachev and V. Lunts, A character formula for the representation of the Weyl group in the cohomology of the associated toric variety Journal of Algebra, 168, 741-772, (1994).
Thomas Godland and Zakhar Kabluchko, Projections and angle sums of permutohedra and other polytopes, arXiv:2009.04186 [math.MG], 2020.
Thomas Godland and Zakhar Kabluchko, Projections and Angle Sums of Belt Polytopes and Permutohedra, Res. Math. (2023) Vol. 78, Art. No. 140.
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 pp. 8-9.
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207v5 [math.CO], 2005-2006.
Shi-Mei Ma, T. Mansour, and D. Callan, Some combinatorial arrays related to the Lotka-Volterra system, arXiv:1404.0731 [math.CO], 2014.
E. Munarini, Characteristic, admittance and matching polynomials of an antiregular graph, Appl. Anal. Discrete Math 3 (1) (2009) 157-176.
Tillmann Nett, Nadine Nett and Andreas Glöckner, Bayesian Analysis of Processed Information in Decision Making Experiments, FernUniversität (Hagen, Germany), University of Cologne (Germany, 2019).
Tilman Piesk, Sets of fixed points of permutations of the n-cube: n = 3 and 4.
Bruce E. Sagan and Joshua P. Swanson, q-Stirling numbers in type B, arXiv:2205.14078 [math.CO], 2022.
Ruedi Suter, Two analogues of a classical sequence, J. Integer Sequences, Vol. 3 (2000), #P00.1.8.
D. G. L. Wang, The Limiting Distribution of the Number of Block Pairs in Type B Set Partitions, arXiv:1108.1264 [math.CO], 2011.

Cf. A154537, A214406, A039756, A039757, A122848, A008277, A048993, A007405 (row sums).

[[(&+[(-1)^(k-j)*(2*j+1)^n*Binomial(k, j): j in [0..k]])/( 2^k*Factorial(k)): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Feb 14 2019

A039755 := proc(n,k) if k < 0 or k > n then 0 ; elif n <= 1 then 1; else procname(n-1,k-1)+(2*k+1)*procname(n-1,k) ; end if; end proc:
seq(seq(A039755(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Oct 30 2009

t[n_, k_] = Sum[(-1)^(k-j)*(2j+1)^n*Binomial[k, j], {j, 0, k}]/(2^k*k!); Flatten[Table[t[n, k], {n, 0, 10}, {k, 0, n}]][[1 ;; 56]]
(* Jean-François Alcover, Jun 09 2011, after Peter Bala *)

T(n,k)=if(k<0 || k>n,0,n!*polcoeff(polcoeff(exp(x+y/2*(exp(2*x+x*O(x^n))-1)),n),k))

[[sum((-1)^(k-j)*(2*j+1)^n*binomial(k, j) for j in (0..k))/( 2^k*factorial(k)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Feb 14 2019

E.g.f. row polynomials: exp(x + y/2 * (exp(2*x) - 1)).
T(n,k) = T(n-1,k-1) + (2*k+1)*T(n-1,k) with T(0,k) = 1 if k=0 and 0 otherwise. Sum_{k=0..n} T(n,k) = A007405(n). - R. J. Mathar, Oct 30 2009; corrected by Joshua Swanson, Feb 14 2019
T(n,k) = (1/(2^k*k!)) * Sum_{j=0..k} (-1)^(k-j)*C(k,j)*(2*j+1)^n.
T(n,k) = (1/(2^k*k!)) * A145901(n,k). - Peter Bala
The row polynomials R(n,x) satisfy the Dobinski-type identity:
R(n,x) = exp(-x/2)*Sum_{k >= 0} (2*k+1)^n*(x/2)^k/k!, as well as the recurrence equation R(n+1,x) = (1+x)*R(n,x)+2*x*R'(n,x). The polynomial R(n,x) has all real zeros (apply [Liu et al., Theorem 1.1] with f(x) = R(n,x) and g(x) = R'(n,x)). The polynomials R(n,2*x) are the row polynomials of A154537. - Peter Bala, Oct 28 2011
Let f(x) = exp((1/2)*exp(2*x)+x). Then the row polynomials R(n,x) are given by R(n,exp(2*x)) = (1/f(x))*(d/dx)^n(f(x)). Similar formulas hold for A008277, A105794, A111577, A143494 and A154537. - Peter Bala, Mar 01 2012
From Peter Bala, Jul 20 2012: (Start)
The o.g.f. for the n-th diagonal (with interpolated zeros) is the rational function D^n(x), where D is the operator x/(1-x^2)*d/dx. For example, D^3(x) = x*(1+8*x^2+3*x^4)/(1-x^2)^5 = x + 13*x^3 + 58*x^5 + 170*x^7 + ... . See A214406 for further details.
An alternative formula for the o.g.f. of the n-th diagonal is exp(-x/2)*(Sum_{k >= 0} (2*k+1)^(k+n-1)*(x/2*exp(-x))^k/k!).
(End)
From Tom Copeland, Dec 31 2015: (Start)
T(n,m) = Sum_{i=0..n-m} 2^(n-m-i)*binomial(n,i)*St2(n-i,m), where St2(n,k) are the Stirling numbers of the second kind, A048993 (also A008277). See p. 755 of Dolgachev and Lunts.
The relation of this entry's e.g.f. above to that of the Bell polynomials, Bell_n(y), of A048993 establishes this formula from a binomial transform of the normalized Bell polynomials, NB_n(y) = 2^n Bell_n(y/2); that is, e^x exp[(y/2)(e^(2x)-1)] = e^x exp[x*2*Bell.(y/2)] = exp[x(1+NB.(y))] = exp(x*P.(y)), so the row polynomials of this entry are given by P_n(y) = [1+NB.(y)]^n = Sum_{k=0..n} C(n,k) NB_k(y) = Sum_{k=0..n} 2^k C(n,k) Bell_k(y/2).
The umbral compositional inverses of the Bell polynomials are the falling factorials Fct_n(y) = y! / (y-n)!; i.e., Bell_n(Fct.(y)) = y^n = Fct_n(Bell.(y)). Since P_n(y) = [1+2Bell.(y/2)]^n, the umbral inverses are determined by [1 + 2 Bell.[ 2 Fct.[(y-1)/2] / 2 ] ]^n = [1 + 2 Bell.[ Fct.[(y-1)/2] ] ]^n = [1+y-1]^n = y^n. Therefore, the umbral inverse sequence of this entry's row polynomials is the sequence IP_n( y) = 2^n Fct_n[(y-1)/2] = (y-1)(y-3) .. (y-2n+1) with IP_0(y) = 1 and, from the binomial theorem, with e.g.f. exp[x IP.(y)]= exp[ x 2Fct.[(y-1)/2] ] = (1+2x)^[(y-1)/2] = exp[ [(y-1)/2] log(1+2x) ].
(End)
Let B(n,k) = T(n,k)*((2*k)!)/(2^k*k!) and P(n,x) = Sum_{k=0..n} B(n,k)*x^(2*k+1). Then (1) P(n+1,x) = (x+x^3)*P'(n,x) for n >= 0, and (2) Sum_{n>=0} B(n,k)/(n!)*t^n = binomial(2*k,k)*exp(t)*(exp(2*t)-1)^k/4^k for k >= 0, and (3) Sum_{n>=0} t^n* P(n,x)/(n!) = x*exp(t)/sqrt(1+x^2-x^2*exp(2*t)). - Werner Schulte, Dec 12 2016
From Wolfdieter Lang, May 26 2017: (Start)
G.f. column k: x^k/Product_{j=0..k} (1 - (1+2*j)*x), k >= 0.
T(n, k) = h^{(k+1)}_{n-k}, the complete homogeneous symmetric function of degree n-k of the k+1 symbols a_j = 1 + 2*j, j = 0, 1, ..., k. (End)
With p(n, x) = Sum_{k=0..n} A001147(k) * T(n, k) * x^k for n >= 0 holds:
  (1) Sum_{i=0..n} p(i, x)*p(n-i, x) = 2^n*(Sum_{k=0..n} A028246(n+1, k+1)*x^k);
  (2) p(n, -1/2) = (n!) * ([t^n] sqrt(2 / (1 + exp(-2*t)))). - Werner Schulte, Feb 16 2024

A049385 Triangle of numbers related to triangle A049375; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297...

Original entry on oeis.org

1, 6, 1, 66, 18, 1, 1056, 372, 36, 1, 22176, 9240, 1200, 60, 1, 576576, 271656, 42840, 2940, 90, 1, 17873856, 9269568, 1685376, 142800, 6090, 126, 1, 643458816, 360847872, 73313856, 7254576, 386400, 11256, 168, 1, 26381811456, 15799069440

Offset: 1

Wolfdieter Lang

a(n,m) := S2(6; n,m) is the sixth triangle of numbers in the sequence S2(k; n,m), k=1..6: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, respectively. a(n,1)= A008548(n).

a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 6-ary trees. Proof based on the a(n,m) recurrence. See also the F. Bergeron et al. reference, especially Table 1, first row and Example 1 for the e.g.f. for m=1. - Wolfdieter Lang, Sep 14 2007

			Triangle begins:
  {1};
  {6,1};
  {66,18,1};
  {1056,372,36,1};
  ...

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48. Added Mar 01 2014.
F. Bergeron, Philippe Flajolet and Bruno Salvy, Varieties of increasing trees, HAL, Rapport De Recherche Inria. Added Mar 01 2014.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
W. Lang, First 10 rows.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO]. - From _N. J. A. Sloane_, Aug 21 2012
E. Neuwirth, Recursively defined combinatorial Functions: Extending Galton's board, Discr. Maths. 239 (2001) 33-51.

Cf. A049412.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> mul(5*k+1, k=0..n), 9); # Peter Luschny, Jan 28 2016

a[n_, m_] := n!*Coefficient[Series[((-1 + (1 - 5*x)^(-1/5))^m)/m!, {x, 0, n}], x^n];
Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 38]]
(* Jean-François Alcover, Jun 21 2011, after e.g.f. *)
rows = 9;
t = Table[Product[5k+1, {k, 0, n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

a(n, m) = n!*A049375(n, m)/(m!*5^(n-m)); a(n+1, m) = (5*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n

a(n, m) = sum(|A051150(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. to Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.

A223168 Triangle S(n, k) by rows: coefficients of 2^((n-1)/2)(x^(1/2)d/dx)^n when n is odd, and of 2^(n/2)(x^(1/2)d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 12, 4, 15, 20, 4, 15, 90, 60, 8, 105, 210, 84, 8, 105, 840, 840, 224, 16, 945, 2520, 1512, 288, 16, 945, 9450, 12600, 5040, 720, 32, 10395, 34650, 27720, 7920, 880, 32, 10395, 124740, 207900, 110880, 23760, 2112, 64, 135135, 540540, 540540, 205920, 34320, 2496, 64

Offset: 0

Udita Katugampola, Mar 17 2013

Also coefficients in the expansion of k-th derivative of exp(n*x^2), see Mathematica program. - Vaclav Kotesovec, Jul 16 2013

			Triangle begins:
       1;
       1,      2;
       3,      2;
       3,     12,      4;
      15,     20,      4;
      15,     90,     60,      8;
     105,    210,     84,      8;
     105,    840,    840,    224,    16;
     945,   2520,   1512,    288,    16;
     945,   9450,  12600,   5040,   720,   32;
   10395,  34650,  27720,   7920,   880,   32;
   10395, 124740, 207900, 110880, 23760, 2112, 64;
  135135, 540540, 540540, 205920, 34320, 2496, 64;
  .
Expansion takes the form:
2^0 (x^(1/2)*d/dx)^1 = 1*x^(1/2)*d/dx.
2^1 (x^(1/2)*d/dx)^2 = 1*d/dx + 2*x*d^2/dx^2.
2^1 (x^(1/2)*d/dx)^3 = 3*x^(1/2)*d^2/dx^2 + 2*x^(3/2)*d^3/dx^3.
2^2 (x^(1/2)*d/dx)^4 = 3*d^2/dx^2 + 12*x*d^3/dx^3 + 4*x^2*d^4/dx^4.
2^2 (x^(1/2)*d/dx)^5 = 15*x^(1/2)*d^3/dx^3 + 20*x^(3/2)*d^4/dx^4 + 4*x^(5/2)*d^5/dx^5.
`
`

Vincenzo Librandi, Rows n = 0..60, flattened
U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229 [math.CA], 2014.

Odd rows includes absolute values of A098503 from right to left.

Cf. A223169-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

a[0]:= f(x);
for i from 1 to 13 do
a[i]:= simplify(2^((i+1)mod 2)*x^(1/2)*(diff(a[i-1],x$1)));
end do;

Flatten[CoefficientList[Expand[FullSimplify[Table[D[E^(n*x^2),{x,k}]/(E^(n*x^2)*(2*n)^Floor[(k+1)/2]),{k,1,13}]]]/.x->1,n]] (* Vaclav Kotesovec, Jul 16 2013 *)

A223172 Triangle S(n,k) by rows: coefficients of 6^((n-1)/2)(x^(1/6)d/dx)^n when n is odd, and of 6^(n/2)(x^(5/6)d/dx)^n when n is even.

Original entry on oeis.org

1, 1, 6, 7, 6, 7, 84, 36, 91, 156, 36, 91, 1638, 1404, 216, 1729, 4446, 2052, 216, 1729, 41496, 53352, 16416, 1296, 43225, 148200, 102600, 21600, 1296, 43225, 1296750, 2223000, 1026000, 162000, 7776, 1339975, 5742750, 5301000, 1674000, 200880, 7776

Offset: 0

Udita Katugampola, Mar 20 2013

			Triangle begins:
        1;
        1,        6;
        7,        6;
        7,       84,        36;
       91,      156,        36;
       91,     1638,      1404,      216;
     1729,     4446,      2052,      216;
     1729,    41496,     53352,    16416,     1296;
    43225,   148200,    102600,    21600,     1296;
    43225,  1296750,   2223000,  1026000,   162000,    7776;
  1339975,  5742750,   5301000,  1674000,   200880,    7776;
  1339975, 48239100, 103369500, 63612000, 15066000, 1446336, 46656;

U. N. Katugampola, Mellin Transforms of Generalized Fractional Integrals and Derivatives, Appl. Math. Comput. 257(2015) 566-580.
U. N. Katugampola, Existence and Uniqueness results for a class of Generalized Fractional Differential Equations, arXiv preprint arXiv:1411.5229 [math.CA], 2014.

Cf. A223168-A223172, A223523-A223532, A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223511-A223522.

a[0]:= f(x):
for i from 1 to 13 do
a[i] := simplify(6^((i+1)mod 2)*x^((4((i+1)mod 2)+1)/6)*(diff(a[i-1],x$1 )));
end do;

A092082 Triangle of numbers related to triangle A092083; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...

Original entry on oeis.org

1, 7, 1, 91, 21, 1, 1729, 511, 42, 1, 43225, 15015, 1645, 70, 1, 1339975, 523705, 69300, 4025, 105, 1, 49579075, 21240765, 3226405, 230300, 8330, 147, 1, 2131900225, 984172735, 166428990, 13820205, 621810, 15386, 196, 1, 104463111025

Offset: 1

Wolfdieter Lang, Mar 19 2004

a(n,m) := S2(7; n,m) is the seventh triangle of numbers in the sequence S2(k;n,m), k=1..6: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, A049385, respectively. a(n,1)=A008542(n), n>=1.
a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 7-ary trees. Proof based on the a(n,m) recurrence. See also the F. Bergeron et al. reference, especially Table 1, first row and Example 1 for the e.g.f. for m=1. - Wolfdieter Lang, Sep 14 2007
Also the Bell transform of A008542(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			{1}; {7,1}; {91,21,1}; {1729,511,42,1}; ...

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, (1992), pp. 24-48.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem., arXiv:quant-phys/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
W. Lang, First 10 rows.
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. - From N. J. A. Sloane, Aug 21 2012

Cf. A092084 (row sums), A092085 (alternating row sums).

# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> mul(6*k+1, k=0..n), 9); # Peter Luschny, Jan 26 2016

mmax = 9; a[n_, m_] := n!*Coefficient[Series[((-1 + (1 - 6*x)^(-1/6))^m)/m!, {x, 0, mmax}], x^n];
Flatten[Table[a[n, m], {n, 1, mmax}, {m, 1, n}]][[1 ;; 37]] (* Jean-François Alcover, Jun 22 2011, after e.g.f. *)
rows = 9;
t = Table[Product[6k+1, {k, 0, n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

a(n, m) = sum(|A051151(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m) := A008277(j, m) (Stirling2 triangle). Priv. comm. with Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.

a(n, m) = n!*A092083(n, m)/(m!*6^(n-m)); a(n+1, m) = (6*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n

E.g.f. for m-th column: ((-1+(1-6*x)^(-1/6))^m)/m!.

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A008297 Triangle of Lah numbers.

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A035469 Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.

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A223168 Triangle S(n, k) by rows: coefficients of 2^((n-1)/2)(x^(1/2)d/dx)^n when n is odd, and of 2^(n/2)(x^(1/2)d/dx)^n when n is even.

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