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A259759 Triangle read by rows: T(n,k) = number of partial idempotent mappings (of an n-chain) with collapse exactly k.

%I A259759 #27 Mar 24 2017 00:47:58
%S A259759 1,2,0,4,0,2,8,0,12,3,16,0,48,24,16,32,0,160,120,160,65,64,0,480,480,
%T A259759 960,780,336,128,0,1344,1680,4480,5460,4704,1897,256,0,3584,5376,
%U A259759 17920,29120,37632,30352,11824,512,0,9216,16128,64512,131040,225792,273168,212832,80145
%N A259759 Triangle read by rows: T(n,k) = number of partial idempotent mappings (of an n-chain) with collapse exactly k.
%D A259759 F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).
%F A259759 T(n,k) = binomial(n,k)Sum_{r=k}^n binomial(n-k,r-k)Sum_{j=0}^k binomial(k,j)S(k-j,j)j!, where S (x,y) is the Stirling numbers of the second kind, which gives the number of ways to partition x into y nonempty subsets.
%e A259759 T (3,2) = 12 because there are exactly 12 partial idempotent mappings (of a 3-chain) with collapse exactly 2, namely: (123-->113), (123-->121), (123-->122), (123-->223), (123-->133), (123--> 323), (12-->11), (12-->22), (23-->22), (23-->33), (13-->11), (13-->33).
%e A259759 Triangle starts:
%e A259759 1;
%e A259759 2,0;
%e A259759 4,0,2;
%e A259759 8,0,12,3;
%e A259759 16,0,48,24,16;
%e A259759 ...
%o A259759 (PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(n,k)*sum(r=k, n, binomial(n-k,r-k)*sum(j=0, k, binomial(k,j)*stirling(k-j,j,2)*j!)), ", ");); print(););} \\ _Michel Marcus_, Jul 15 2015
%K A259759 nonn,tabl
%O A259759 0,2
%A A259759 _Wafa AlNadabi_, Jul 04 2015
%E A259759 More terms from _Michel Marcus_, Jul 15 2015