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%I #42 Aug 27 2015 19:19:45
%S 1,1,1,1,2,3,1,4,7,11,1,8,17,30,48,1,16,43,86,150,241,1,32,113,258,
%T 492,846,1358,1,64,307,806,1686,3108,5276,8445,1,128,857,2610,6012,
%U 11904,21392,35904,57256,1,256,2443,8726,22230,47376,90224,158880,263976,419233
%N Triangle read by rows: T(n,k) = number of partial idempotent mappings (of an n-chain) with (right) waist exactly k.
%D F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).
%F T(n,k) = Sum_{m=0..k} binomial(k-1,m-1) * (m+1)^(n-m).
%e T(3,2) = 7 because there are exactly 7 partial idempotent mappings (of a 3-chain) with right waist exactly 2, namely: (123-->222), (123-->122), (123-->121), (12-->22), (12-->12), (23-->22), (2-->2).
%e Triangle starts:
%e 1;
%e 1,1;
%e 1,2,3;
%e 1,4,7,11;
%e 1,8,17,30,48;
%e ...
%o (PARI) mybinom(x,y) = if ((x==-1) && (y==-1), 1, binomial(x,y));
%o tabl(nn) = {for (n=0, nn, for (k=0, n, print1(sum(m=0, k, mybinom(k-1, m-1) * (m+1)^(n-m)), ", "); ); print(); ); } \\ _Michel Marcus_, Jul 15 2015
%K nonn,tabl
%O 0,5
%A _Wafa AlNadabi_, Jul 04 2015
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