A131222 Exponential Riordan array [1, log((1-x)/(1-2x))].

1, 0, 1, 0, 3, 1, 0, 14, 9, 1, 0, 90, 83, 18, 1, 0, 744, 870, 275, 30, 1, 0, 7560, 10474, 4275, 685, 45, 1, 0, 91440, 143892, 70924, 14805, 1435, 63, 1, 0, 1285200, 2233356, 1274196, 324289, 41160, 2674, 84, 1

Offset: 0

Author

Paul Barry, Jun 18 2007

Keywords

• easy
• nonn
• tabl

Comments

This is also the matrix product of the unsigned Lah numbers and the Stirling cycle numbers. See also A079639 and A079640 for variants based on an (1,1)-offset of the number triangles. - Peter Luschny, Apr 12 2015

The Bell transform of A029767(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

Essentially the same as A079638. - Peter Bala, Feb 12 2022

Examples

			Number triangle starts:
  1,
  0,   1;
  0,   3,   1;
  0,  14,   9,   1;
  0,  90,  83,  18,  1;
  0, 744, 870, 275, 30,  1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Paul Barry, Exponential Riordan arrays and permutation enumeration,Journal of Integer Sequences, Vol. 13 (2010).

Crossrefs

Cf. A000007, A002866, A029767, A079638, A079639, A079640.

Programs

Maple

RioExp := (d,h,n,k) -> coeftayl(d*h^k, x=0,n)*n!/k!:
A131222 := (n,k) -> RioExp(1,log((1-x)/(1-2*x)),n,k):
seq(print(seq(A131222(n,k),k=0..n)),n=0..5); # Peter Luschny, Apr 15 2015
# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n=0,1,n!*(2^(n+1)-1)), 9); # Peter Luschny, Jan 27 2016

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 = 12;
M = BellMatrix[If[# == 0, 1, #! (2^(#+1) - 1)]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

Maxima

T(n,m):=if n=0 and m=0 then 1 else n!*sum((stirling1(k,m)*2^(n-k)*binomial(n-1,k-1))/k!,k,m,n); /* Vladimir Kruchinin, Sep 27 2012 */

Sage

def Lah(n, k):
    if n == k: return 1
    if k<0 or  k>n: return 0
    return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))
matrix(ZZ, 8, Lah) * matrix(ZZ, 8, stirling_number1) # as a square matrix Peter Luschny, Apr 12 2015
# alternatively:

Sage

# uses[bell_matrix from A264428]
bell_matrix(lambda n: A029767(n+1), 10) # Peter Luschny, Jan 18 2016

Formula

Row sums are A002866.

Second column is A029767.

T(n,m) = n! * Sum_{k=m..n} Stirling1(k,m)*2^(n-k)*binomial(n-1,k-1)/k!, n >= m >= 0. - Vladimir Kruchinin, Sep 27 2012