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Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).
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%I #30 Nov 28 2018 15:26:03

%S 1,2,1,6,5,1,24,26,9,1,120,154,71,14,1,720,1044,580,155,20,1,5040,

%T 8028,5104,1665,295,27,1,40320,69264,48860,18424,4025,511,35,1,362880,

%U 663696,509004,214676,54649,8624,826,44,1,3628800,6999840,5753736,2655764

%N Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).

%C 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.

%C 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

%C 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

%C 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

%H G. C. Greubel, <a href="/A136124/b136124.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Olivier Bodini, Antoine Genitrini, Mehdi Naima, <a href="https://arxiv.org/abs/1808.08376">Ranked Schröder Trees</a>, arXiv:1808.08376 [cs.DS], 2018.

%F 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.

%e T(6,3)=71 because (-1)^9*[s(6,1)+s(6,2)+s(6,3)]=-(-120+274-225)=71.

%e Triangle starts:

%e 1;

%e 2, 1;

%e 6, 5, 1;

%e 24, 26, 9, 1;

%e 120, 154, 71, 14, 1;

%p 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

%p 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

%t 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 *)

%Y Cf. A000142, A008275, A001705, A001706, A001707, A001708, A001710.

%Y Cf. A049444, A143491. [_Peter Bala_, Aug 25 2008]

%K nonn,tabl

%O 2,2

%A _Emeric Deutsch_, Dec 23 2007