A049403 A triangle of numbers related to triangle A030528; array a(n,m), read by rows (1 <= m <= n).

1, 1, 1, 0, 3, 1, 0, 3, 6, 1, 0, 0, 15, 10, 1, 0, 0, 15, 45, 15, 1, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0, 0, 0, 0, 10395, 62370

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn
• tabl

Comments

a(n,1) = A019590(n) = A008279(1,n). a(n,m) =: S1(-1; n,m), a member of a sequence of lower triangular Jabotinsky matrices, including S1(1; n,m) = A008275 (signed Stirling first kind), S1(2; n,m) = A008297(n,m) (signed Lah numbers). a(n,m) matrix is inverse to signed matrix ((-1)^(n-m))*A001497(n-1,m-1) (signed Bessel triangle). The monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m, E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).

Exponential Riordan array [1+x, x(1+x/2)]. T(n,k) = A001498(k+1, n-k). - Paul Barry, Jan 15 2009

Examples

			Triangle a(n,m) (with rows n >= 1 and columns m >= 1) begins as follows:
  1;                 with row polynomial E(1,x) = x;
  1, 1;              with row polynomial E(2,x) = x^2 + x;
  0, 3,  1;          with row polynomial E(3,x) = 3*x^2 + x^3;
  0, 3,  6,   1;     with row polynomial E(4,x) = 3*x^2 + 6*x^3 + x^4;
  0, 0, 15,  10,   1;
  0, 0, 15,  45,  15,   1;
  0, 0,  0, 105, 105,  21,  1;
  0, 0,  0, 105, 420, 210, 28, 1;
  ...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
• Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
• Wolfdieter Lang, First 10 rows of the array and more.
• Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.

Crossrefs

Cf. A000085 (row sums), A001497, A001498, A008275, A008279, A008297, A019590, A030528, A039692.

Variations of this array: A096713, A104556, A122848, A130757.

Programs

Maple

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n<2,1,0), 9); # Peter Luschny, Jan 28 2016

Mathematica

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 11}, {k, n}] // Flatten
(* Second program: *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 13;
M = BellMatrix[If[#<2, 1, 0]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

Formula

a(n, m) = n!*A030528(n, m)/(m!*2^(n-m)) for n >= m >= 1.

a(n, m) = (2*m-n+1)*a(n-1, m) + a(n-1, m-1) for n >= m >= 1 with a(n, m) = 0 for n < m, a(n, 0) := 0, and a(1, 1) = 1. [The 0th column does not appear in this array. - Petros Hadjicostas, Oct 28 2019]

E.g.f. for the m-th column: (x*(1 + x/2))^m/m!.

a(n,m) = A122848(n,m). - R. J. Mathar, Jan 14 2011