OFFSET
2,2
COMMENTS
Sum of entries in row n = n!/2 = A001710(n). T(n,1) = (n-1)! = A000142(n-1). Columns 2,3,4 and 5 yield A001705,A001706,A001707 and A001708, respectively.
See A143491 for the interpretation of these numbers as restricted Stirling numbers of the first kind. See A049444 for a signed version of this array. - Peter Bala, Aug 25 2008
With offset n=0, k=0: triangle T(n,k), read by rows, given by [2,1,3,2,4,3,5,4,6,5,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 29 2011
With offset n=0, k=0: T(n,k) is the number of ways to seat n people at any number of round tables and serve exactly k of the tables water, some number of the remaining tables red wine, and the rest of the tables white wine. - Geoffrey Critzer, Mar 13 2015
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Olivier Bodini, Antoine Genitrini, Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.
FORMULA
E.g.f.: Sum[(1/n!)T(n,k)x^n*t^k, k=1..n-1, n>=2]=1/[(1+t)(1-x)^t]-(1+tx)/(1+t). Generating polynomial of row n = t*Product(j+t, j=2..n-1). T(n,k) is the sum of all products of n-k-1 different integers taken from {2,3,...,n-1}. For example, T(6,3) = 2*3 + 2*4 + 2*5 + 3*4 + 3*5 + 4*5 = 71.
EXAMPLE
T(6,3)=71 because (-1)^9*[s(6,1)+s(6,2)+s(6,3)]=-(-120+274-225)=71.
Triangle starts:
1;
2, 1;
6, 5, 1;
24, 26, 9, 1;
120, 154, 71, 14, 1;
MAPLE
A136124_row := proc(n) local k, j; `if`(n=0, 1, seq((-1)^(n+1-k)*add(stirling1(n+1, j), j=1..k), k=1..n)) end: seq(print(A136124_row(r)), r=1..6); # Peter Luschny, Sep 29 2011
with(combinat): T:=proc(n, k) options operator, arrow: (-1)^(n+k)*(sum(stirling1(n, j), j=1..k)) end proc: for n from 2 to 11 do seq(T(n, k), k=1..n-1) end do; # yields sequence in triangular form
MATHEMATICA
nn = 10; Map[Select[#, # > 0 &] &, Range[0, nn]!CoefficientList[Series[Exp[(2 + y) Log[1/(1 - x)]], {x, 0, nn}], {x, y}]] // Flatten (* Geoffrey Critzer, Mar 13 2015 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Dec 23 2007
STATUS
approved