1,2

Previous name was: A triangle of numbers related to the triangle A035324; generalization of Stirling numbers of second kind A008277 and Lah numbers A008297.

If one replaces in the recurrence the '2' by '0', resp. '1', one obtains the Lah-number, resp. Stirling-number of 2nd kind, triangle A008297, resp. A008277.

The product of two lower triangular Jabotinsky matrices (see A039692 for the Knuth 1992 reference) is again such a Jabotinsky matrix: J(n,m) = Sum_{j=m..n} J1(n,j)*J2(j,m). The e.g.f.s of the first columns of these triangular matrices are composed in the reversed order: f(x)=f2(f1(x)). With f1(x)=-(log(1-2*x))/2 for J1(n,m)=|A039683(n,m)| and f2(x)=exp(x)-1 for J2(n,m)=A008277(n,m) one has therefore f2(f1(x))=1/sqrt(1-2*x) - 1 = f(x) for J(n,m)=a(n,m). This proves the matrix product given below. The m-th column of a Jabotinsky matrix J(n,m) has e.g.f. (f(x)^m)/m!, m>=1.

a(n,m) gives the number of forests with m rooted ordered trees with n non-root vertices labeled in an organic way. Organic labeling means that the vertex labels along the (unique) path from the root with label 0 to any leaf (non-root vertex of degree 1) is increasing. Proof: first for m=1 then for m>=2 using the recurrence relation for a(n,m) given below. - Wolfdieter Lang, Aug 07 2007

Also the Bell transform of A001147(n+1) (adding 1,0,0,... as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

If D is the operator d/dx, and E is the operator (x^3)d/dx, then we have E^n = Sum_{k=1..n} T(n,k)*x^(2*n+k)*D^k. - Magus K. Chu, Dec 08 2025

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Peter Bala, Generalized Dobinski formulas

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], 2013-2014.

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

Pawel Blasiak, Karol A. Penson, and Allan 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, 2015.

Tom Copeland, Addendum to Mathemagical Forests, 2010.

Tom Copeland, Mathemagical Forests, 2008.

Askar Dzhumadildaev and Damir Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - N. J. A. Sloane, Mar 28 2015]

Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.

Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) #09.8.3.

Donald E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992; Mathematica J. 2.1 (1992), no. 4, 67-78.

Wolfdieter Lang, On generalizations of Stirling number triangles, J. Int. Seq. 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.

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

Erich Neuwirth, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. (2001) Vol. 239, 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.

Umesh Shankar, Log-concavity of rows of triangular arrays satisfying a certain super-recurrence, arXiv:2508.12467 [math.CO], 2025. See p. 4.

Hanzhong Zhang, Weak-Field Expansion: A Time-Closed Solution of Quantum Three-Wave Mixing, arXiv:2603.29869 [quant-ph], 2026. See pp. 11, 16.

a(n, m) = Sum_{j=m..n} |A039683(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 comment on products of Jabotinsky matrices.

a(n, m) = n!*A035324(n, m)/(m!*2^(n-m)), n >= m >= 1; a(n+1, m)= (2*n+m)*a(n, m)+a(n, m-1); a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1.

E.g.f. of m-th column: ((x*c(x/2)/sqrt(1-2*x))^m)/m!, where c(x) = g.f. for Catalan numbers A000108.

From Vladimir Kruchinin, Mar 30 2011: (Start)

G.f. (1/sqrt(1-2*x) - 1)^k = Sum_{n>=k} (k!/n!)*a(n,k)*x^n.

a(n,k) = 2^(n+k) * n! / (4^n*n*k!) * Sum_{j=0..n-k} (j+k) * 2^(j) * binomial(j+k-1,k-1) * binomial(2*n-j-k-1,n-1). (End)

From Peter Bala, Nov 25 2011: (Start)

E.g.f.: G(x,t) = exp(t*A(x)) = 1 + t*x + (3*t + t^2)*x^2/2! + (15*t + 9*t^2 + t^3)*x^3/3! + ..., where A(x) = -1 + 1/sqrt(1-2*x) satisfies the autonomous differential equation A'(x) = (1+A(x))^3.

The generating function G(x,t) satisfies the partial differential equation t*(dG/dt+G) = (1-2*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)^3*d/dx. Cf. A008277 (D = (1+x)*d/dx), A105278 (D = (1+x)^2*d/dx), A035469 (D = (1+x)^4*d/dx) and A049029 (D = (1+x)^5*d/dx). (End)

The n-th row polynomial R(n,x) is given by the Dobinski-type formula R(n,x) = exp(-x)*Sum_{k>=1} k*(k+2)*...*(k+2*n-2)*x^k/k!. - Peter Bala, Jun 22 2014

T(n,k) = 2^(k-n)*hypergeom([k-n,k+1],[k-2*n+1],2)*Gamma(2*n-k)/(Gamma(k)*Gamma(n-k+1)). - Peter Luschny, Mar 31 2015

T(n,k) = 2^n*Sum_{j=1..k} ((-1)^(k-j)*binomial(k, j)*Pochhammer(j/2, n)) / k!. - Peter Luschny, Mar 04 2024

T(n,k) = T(n-1,k-1)+(2*(n-1)+k)*T(n-1,k). - Magus K. Chu, Dec 08 2025

Matrix begins:

1;

3, 1;

15, 9, 1;

105, 87, 18, 1;

945, 975, 285, 30, 1;

...

Combinatoric meaning of a(3,2)=9: The nine increasing path sequences for the three rooted ordered trees with leaves labeled with 1,2,3 and the root labels 0 are: {(0,3),[(0,1),(0,2)]}; {(0,3),[(0,2),(0,1)]}; {(0,3),(0,1,2)}; {(0,1),[(0,3),(0,2)]}; [(0,1),[(0,2),(0,3)]]; [(0,2),[(0,1),(0,3)]]; {(0,2),[(0,3),(0,1)]}; {(0,1),(0,2,3)}; {(0,2),(0,1,3)}.

T := (n, k) -> 2^(k-n)*hypergeom([k-n, k+1], [k-2*n+1], 2)*GAMMA(2*n-k)/

(GAMMA(k)*GAMMA(n-k+1)); for n from 1 to 9 do seq(simplify(T(n, k)), k=1..n) od; # Peter Luschny, Mar 31 2015

# Alternative:

T := (n, k) -> local j; 2^n*add((-1)^(k-j)*binomial(k, j)*pochhammer(j/2, n), j = 1..k)/k!: for n from 1 to 6 do seq(T(n, k), k=1..n) od; # Peter Luschny, Mar 04 2024

a[n_, k_] := 2^(n+k)*n!/(4^n*n*k!)*Sum[(j+k)*2^(j)*Binomial[j + k - 1, k-1]*Binomial[2*n - j - k - 1, n-1], {j, 0, n-k}]; Flatten[Table[a[n, k], {n, 1, 9}, {k, 1, n}] ] [[1 ;; 40]] (* Jean-François Alcover, Jun 01 2011, after Vladimir Kruchinin *)

(Maxima) a(n, k):=2^(n+k)*n!/(4^n*n*k!)*sum((j+k)*2^(j)*binomial(j+k-1, k-1)*binomial(2*n-j-k-1, n-1), j, 0, n-k); /* Vladimir Kruchinin, Mar 30 2011 */

(Haskell)

a035342 n k = a035342_tabl !! (n-1) !! (k-1)

a035342_row n = a035342_tabl !! (n-1)

a035342_tabl = map fst $ iterate (\(xs, i) -> (zipWith (+)

([0] ++ xs) $ zipWith (*) [i..] (xs ++ [0]), i + 2)) ([1], 3)

-- Reinhard Zumkeller, Mar 12 2014

(SageMath) # uses[bell_matrix from A264428]

# Adds a column 1, 0, 0, 0, ... at the left side of the triangle.

print(bell_matrix(lambda n: A001147(n+1), 9)) # Peter Luschny, Jan 19 2016

The column sequences are A001147, A035101, A035119, ...

Row sums: A049118(n), n >= 1.

Cf. A000108, A035324, A008277, A008297, A094638.

Sequence in context: A282629 A135896 A134144 * A039815 A318392 A329059

Adjacent sequences: A035339 A035340 A035341 * A035343 A035344 A035345

easy,nice,nonn,tabl,changed

Simpler name from Peter Luschny, Mar 31 2015

approved