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A226780 Triangular array read by rows. T(n,k) is the number of 2 tuple lists of length n that have exactly k coincidences; n >= 0, 0 <= k <= n.

%I A226780 #13 Jun 21 2013 20:08:06
%S A226780 1,0,1,3,0,1,26,9,0,1,453,104,18,0,1,11844,2265,260,30,0,1,439975,
%T A226780 71064,6795,520,45,0,1,22056222,3079825,248724,15855,910,63,0,1,
%U A226780 1436236809,176449776,12319300,663264,31710,1456,84,0,1
%N A226780 Triangular array read by rows. T(n,k) is the number of 2 tuple lists of length n that have exactly k coincidences; n >= 0, 0 <= k <= n.
%C A226780 Consider the set (with cardinality n!^2) of (ordered) lists of n two tuples such that all numbers from 1 to n appear as the first as well as the second tuple entry.  If the j-th two tuple of the list is (j,j) then call it a coincidence.  T(n,k) is the number of such lists that have k coincidences.
%H A226780 Alois P. Heinz, <a href="/A226780/b226780.txt">Rows n = 0..100, flattened</a>
%F A226780 T(n,k) = binomial(n,k) * A089041(n-k).
%F A226780 Row sums = n!^2.
%F A226780 T(n,0) = A089041(n).
%F A226780 The expected number of coincidences, Sum_{k=0..n} T(n,k)*k/n!^2 = 1/n.
%e A226780 1;
%e A226780 0,        1;
%e A226780 3,        0,       1;
%e A226780 26,       9,       0,      1;
%e A226780 453,      104,     18,     0,     1;
%e A226780 11844,    2265,    260,    30,    0,   1;
%e A226780 439975,   71064,   6795,   520,   45,  0,   1;
%e A226780 22056222, 3079825, 248724, 15855, 910, 63,  0,   1;
%p A226780 b:= proc(n) option remember;
%p A226780        `if`(n<2, 1-n, n^2*b(n-1)+n*(n-1)*b(n-2)+(-1)^n)
%p A226780     end:
%p A226780 T:= (n, k)-> binomial(n, k) * b(n-k):
%p A226780 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Jun 21 2013
%t A226780 a = Table[Sum[(-1)^k Binomial[n,k](n-k)!^2, {k,0,n}], {n,0,15}]; Table[Drop[Transpose[Table[Table[Binomial[n,i]*a[[n-i+1]], {n,0,10}], {i,0,10}]][[j]], -11+j], {j, 10}]//Grid
%Y A226780 Cf. A008290.
%K A226780 nonn,tabl
%O A226780 0,4
%A A226780 _Geoffrey Critzer_, Jun 18 2013