OFFSET
0,2
COMMENTS
T(n,k) is the number of length k+1 sequences of nonempty mutually disjoint subsets of {1,2,...,n+1}. The e.g.f. for the column corresponding to k is exp(x)*(exp(x)-1)^(k+1). - Geoffrey Critzer, Dec 20 2011
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
F. Brenti and V. Welker, f-vectors of barycentric subdivisions Math. Z., 259(4), 849-865, 2008.
Wikipedia, Barycentric subdivision
FORMULA
T(0,k) = delta(0,k), T(n,k) = delta(0,k) + (k+1)(T(n-1,k-1) + (k+2)T(n-1,k)).
E.g.f.: exp(x)*(exp(x)-1)/(1-y*(exp(x)-1)). - Vladeta Jovovic, Apr 13 2003
T(n,k) = Sum_{i = 0..n} binomial(n+1,i+1)*(k+1)!*Stirling2(i+1,k+1) = (k+1)!*Stirling2(n+2,k+2) (Brenti and Welker). Row sums are A002050. - Peter Bala, Jul 12 2014
EXAMPLE
T(2,1) = 12 because there are 12 such length 2 sequences of subsets of {1,2,3}: ({1},{2}), ({1},{3}), ({2},{3}), ({1},{2,3}), ({2},{1,3}), ({3},{1,2}) with two orderings for each. - Geoffrey Critzer, Dec 20 2011
Triangle begins:
1
3 2
7 12 6
15 50 60 24
31 180 390 360 120
MAPLE
with(combinat):
a := (n, k) -> (k+1)!*stirling2(n+2, k+2):
seq(print(seq(a(n, k), k = 0..n)), n = 0..10);
MATHEMATICA
nn = 5; a = Exp[ x] - 1 ; f[list_] := Select[list, # > 0 &]; Map[f, Transpose[Table[Drop[Range[0, nn]!CoefficientList[Series[a^k Exp[x], {x, 0, nn}], x], 1], {k, 1, 5}]]] // Grid (* Geoffrey Critzer, Dec 20 2011 *)
Table[(k+1)!*StirlingS2[n+2, k+2], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 19 2017 *)
PROG
(PARI) for(n=0, 10, for(k=0, n, print1((k+1)!*stirling(n+2, k+2, 2), ", "))) \\ G. C. Greubel, Nov 19 2017
CROSSREFS
KEYWORD
AUTHOR
Rob Arthan, Jan 12 2000
EXTENSIONS
More terms from James A. Sellers, Jan 14 2000
STATUS
approved