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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.
0
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
0,5
COMMENTS
Also the Bell transform of A000262(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016
FORMULA
T(n+1,1) = A000262(n).
T(n+1,n) = A045943(n).
Row sums are A084357.
EXAMPLE
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;
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 *)
PROG
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
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.
Sequence in context: A249480 A271704 A307419 * A256893 A359759 A137431
KEYWORD
nonn,tabl,easy
AUTHOR
Peter Luschny, Apr 12 2015
STATUS
approved