A256892 Triangular array read by rows, the matrix product of the unsigned Lah numbers and the Stirling set numbers, T(n,k) for n>=0 and 0<=k<=n.

1, 0, 1, 0, 3, 1, 0, 13, 9, 1, 0, 73, 79, 18, 1, 0, 501, 755, 265, 30, 1, 0, 4051, 7981, 3840, 665, 45, 1, 0, 37633, 93135, 57631, 13580, 1400, 63, 1, 0, 394353, 1192591, 911582, 274141, 38290, 2618, 84, 1, 0, 4596553, 16645431, 15285313, 5633922, 999831, 92358, 4494, 108, 1

Offset: 0

Author

Peter Luschny, Apr 12 2015

Keywords

• nonn
• tabl
• easy

Comments

Also the Bell transform of A000262(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

Examples

			Triangle starts:
1;
0,    1;
0,    3,    1;
0,   13,    9,    1;
0,   73,   79,   18,   1;
0,  501,  755,  265,  30,  1;
0, 4051, 7981, 3840, 665, 45, 1;
		

Crossrefs

See also A088814 and A088729 for variants based on an (1,1)-offset of the number triangles. See A131222 for the product Lah * Stirling-cycle.

A079640 is an unsigned matrix inverse reduced to an (1,1)-offset.

Cf. A000262, A045943, A084357, A088729, A088814.

Programs

Maple

# The function BellMatrix is defined in A264428.
BellMatrix(n -> simplify(hypergeom([-n, -n-1], [], 1)), 9); # Peter Luschny, Jan 29 2016

Mathematica

BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, HypergeometricPFQ[{-n, -n-1}, {}, 1]], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

SageMath

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_number2)  # as a square matrix

Formula

T(n+1,1) = A000262(n).

T(n+1,n) = A045943(n).

Row sums are A084357.