%I A087903
%S A087903 1,1,1,1,4,1,1,11,9,1,1,26,48,16,1,1,57,202,140,25,1,1,120,747,916,325,
%T A087903 36,1,1,247,2559,5071,3045,651,49,1,1,502,8362,25300,23480,8260,1176,64,
%U A087903 1,1,1013,26520,117962,159736,84456,19404,1968,81,1,1,2036,82509,525608
%N A087903 Triangle read by rows of the numbers T(n,k) (n>1, 0<k<n) of set partitions 
               of n of length k which do not have a proper subset of parts with 
               a union equal to a subset {1,2,...,j} with j<n.
%C A087903 T(n,n-1)=T(n,1)=1; T(n,n-2) = (n-2)^2; T(n,2) = A000295(n)
%C A087903 Another version of the triangle T(n,k), 0<=k<=n, given by [1, 0, 2, 0, 
               3, 0, 4, 0, 5, 0, 6, ...] DELTA [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 
               1, ...] where DELTA is the operator defined in A084938; see also 
               A086329 for a triangle transposed . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Jun 13 2004
%D A087903 M. Rosas, B. Sagan, Symmetric functions in noncommuting variables, arXiv:math.CO/
               0208168
%D A087903 M. C. Wolf, Symmetric Functions of Non-commutative Elements, Duke Math. 
               J., 2 (1936), 626-637.
%F A087903 T(n, k) = S2(n-1, k)+sum_{j=0}^{n-2} sum_{d=0}^{k-1} (k-d-1) T(n-j-1, 
               k-d) S2(j, d), where S2(n, k) is the Stirling number of the second 
               kind
%F A087903 Sum_{k = 1, .., n-1} T(n, k) = A074664(n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Jun 13 2004
%F A087903 G.f.: 1-1/(1+add(add(q^n t^k S2(n, k), k=1..n), n >= 1)) where S2(n, 
               k) are the Stirling numbers of the 2nd kind A008277 - Mike Zabrocki 
               (zabrocki(AT)mathstat.yorku.ca), Sep 03 2005
%e A087903 T(2,1)=1 for {12}
%e A087903 T(3,1)=1, T(3,2) = 1 for {123}; {13|2}
%e A087903 T(4,1)=1, T(4,2)=4, T(4,3)=1 for {1234}; {14|23}, {13|24}, {124|3}, {134|2}; 
               {14|2|3}
%e A087903 Comment from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 16 2007: 
               Triangle begins:
%e A087903 1;
%e A087903 1, 1;
%e A087903 1, 4, 1;
%e A087903 1, 11, 9, 1;
%e A087903 1, 26, 48, 16, 1;
%e A087903 1, 57, 202, 140, 25, 1;
%e A087903 1, 120, 747, 916, 325, 36, 1;
%e A087903 1, 247, 2559, 5071, 3045, 651, 49, 1;
%e A087903 1, 502, 8362, 25300, 23480, 8260, 1176, 64, 1;
%e A087903 1, 1013, 26520, 117962, 159736, 84456, 19404, 1968, 81, 1 ;...
%e A087903 Triangle T(n,k), 0<=k<=n, given by [1,0,2,0,3,0,4,0,...] DELTA [0,1,0,
               1,0,1,0,...] begins:
%e A087903 1;
%e A087903 1, 0;
%e A087903 1, 1, 0;
%e A087903 1, 4, 1, 0;
%e A087903 1, 11, 9, 1, 0;
%e A087903 1, 26, 48, 16, 1, 0;
%e A087903 1, 57, 202, 140, 25, 1, 0;
%e A087903 1, 120, 747, 916, 325, 36, 1, 0;
%e A087903 1, 247, 2559, 5071, 3045, 651, 49, 1, 0;
%e A087903 1, 502, 8362, 25300, 23480, 8260, 1176, 64, 1, 0;
%e A087903 1, 1013, 26520, 117962, 159736, 84456, 19404, 1968, 81, 1, 0 ;...
%p A087903 A := proc(n,k) option remember; local j,ell; if n<=0 or k>=n then 0; 
               elif k=1 or k=n-1 then 1; else S2(n-1,k)+add(add((k-ell-1)*A(n-j-1,
               k-ell)*S2(j,ell),ell=0..k-1),j=0..n-2); fi; end: S2 := (n,k)->if 
               n<0 or k>n then 0; elif k=n or k=1 then 1 else k*S2(n-1,k)+S2(n-1,
               k-1); fi:
%Y A087903 Cf. A008277, A055106, A074664, A000110.
%Y A087903 Cf. A055105.
%Y A087903 Sequence in context: A145271 A147564 A090981 this_sequence A112500 A152938 
               A154096
%Y A087903 Adjacent sequences: A087900 A087901 A087902 this_sequence A087904 A087905 
               A087906
%K A087903 easy,nonn,tabl
%O A087903 2,5
%A A087903 Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Oct 14 2003