A038455 Triangle read by rows: T(n, k) = [x^k] x*Pochhammer(n + x, n)/(n + x).

1, 3, 1, 20, 9, 1, 210, 107, 18, 1, 3024, 1650, 335, 30, 1, 55440, 31594, 7155, 805, 45, 1, 1235520, 725592, 176554, 22785, 1645, 63, 1, 32432400, 19471500, 4985316, 705649, 59640, 3010, 84, 1, 980179200, 598482000, 159168428, 24083892, 2267769, 136080, 5082, 108, 1

Offset: 1

Author

Wolfdieter Lang

Keywords

• nonn
• tabl

Comments

Original name: A Jabotinsky-triangle related to A006963.

i) This triangle gives the nonvanishing entries of the Jabotinsky matrix for F(z) = c(z) with c(z) the g.f. for the Catalan numbers A000108. (Notation of F(z) as in Knuth's paper).

ii) E(n, x) = Sum_{m=1..n} a(n, m)*x^m, E(0, x) = 1, are exponential convolution polynomials: E(n, x + y) = Sum_{k=0..n} binomial(n, k)*E(k, x)*E(n-k, y), (cf. Knuth's paper with E(n, x)= n!*F(n, x).)

iii) Explicit formula: see Knuth's paper for f(n, m) formula with f(k) = A006963(k + 1).

Bell polynomial of second kind for log(A000108(x)). - Vladimir Kruchinin, Mar 26 2013

Also the Bell transform of A006963(n+2). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

Examples

			Triangle starts:
  [1]       1;
  [2]       3,      1;
  [3]      20,      9,      1;
  [4]     210,    107,     18,     1;
  [5]    3024,   1650,    335,    30,    1;
  [6]   55440,  31594,   7155,   805,   45,  1;
  [7] 1235520, 725592, 176554, 22785, 1645, 63, 1;
		

Links

• Priyavrat Deshpande and Krishna Menon, A statistic for regions of braid deformations, Séminaire Lotharingien de Combinatoire (2022) Vol. 86, Issue B, Art. No. 23.
• D. E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA], 1992; Mathematica J. 2.1 (1992), no. 4, 67-78.
• J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.

Crossrefs

Cf. A006963, A000108, A001761, A039619, A039646.

Programs

Maple

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (2*n+1)!/(n+1)!, 9); # Peter Luschny, Jan 28 2016
gf := n -> x*pochhammer(n + x, n)/(n + x):
ser := n -> series(gf(n), x, n + 2):
seq(seq(coeff(ser(n), x, k), k = 1..n), n = 1..9);  # Peter Luschny, Jun 27 2024

Mathematica

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 11; M = BellMatrix[(2#+1)!/(#+1)!&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten
(* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

Maxima

a(n,m):=(n-1)!*(sum((stirling1(k,m)*binomial(2*n,n-k))/(k-1)!,k,m,n)); /* Vladimir Kruchinin, Mar 26 2013 */

Formula

a(n, 1) = A006963(n + 1) = (2*n - 1)!/n!, n >= 1;

a(n, m) = Sum_{j=1..n-m+1} binomial(n - 1, j - 1)*A006963(j + 1)*a(n - j, m - 1), n >= m >= 2.

E.g.f.: ((1 - sqrt(1 - 4*x))/(2*x))^y. - Vladeta Jovovic, May 02 2003

a(n, m) = (n - 1)!*(Sum_{k=m..n} Stirling1(k, m)*binomial(2*n, n-k)/(k-1)!). - Vladimir Kruchinin, Mar 26 2013

Extensions

New name by Peter Luschny, Jun 27 2024