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%I #10 Aug 02 2012 06:54:02
%S 1,1,1,1,4,4,1,7,21,21,1,10,47,126,126,1,13,82,324,818,818,1,16,126,
%T 642,2300,5594,5594,1,19,179,1107,4977,16741,39693,39693,1,22,241,
%U 1746,9335,38642,124383,289510,289510,1,25,312,2586,15941,77273,301630,939880,2157150,2157150
%N Triangle enumerating certain two-line arrays of positive integers.
%C This is the table of f(n,k) in the notation of Carlitz (p.123). The triangle enumerates two-line arrays of positive integers
%C ............a_1 a_2 ... a_n..........
%C ............b_1 b_2 ... b_n..........
%C such that
%C 1) max(a_i, b_i) <= min(a_(i+1), b_(i+1)) for 1 <= i <= n-1
%C 2) max(a_i, b_i) <= i for 1 <= i <= n
%C 3) a_n = b_n = k.
%C See A071948 and A193091 for other two-line array enumerations.
%H L. Carlitz, <a href="http://www.fq.math.ca/11-2.html">Enumeration of two-line arrays</a>, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.
%F Recurrence equation:
%F T(1,1) = 1; T(n,n) = T(n,n-1); T(n+1,k) = sum {j = 1..k} (2*k-2*j+1)*T(n,j) for 1 <= k <= n.
%F T(n+1,k+1) = 1/n{(n-k)*sum {i = 0..k} C(n,k-i)*C(2*n+i,i) + sum {i = 1..k} C(n,k-i)*C(2*n+i,i-1)}.
%F Row reverse has production matrix
%F 1 1
%F 3 3 1
%F 5 5 3 1
%F 7 7 5 3 1
%F ...
%F Main diagonal T(n,n) = A003168(n). Row sums A211789.
%e Triangle begins
%e .n\k.|..1....2....3....4....5....6
%e = = = = = = = = = = = = = = = = = =
%e ..1..|..1
%e ..2..|..1....1
%e ..3..|..1....4....4
%e ..4..|..1....7...21...21
%e ..5..|..1...10...47..126..126
%e ..6..|..1...13...82..324..818..818
%e ...
%e T(4,2) = 7: The 7 two-line arrays are
%e ...1 1 1 2....1 1 2 2....1 2 2 2....1 1 1 2
%e ...1 1 1 2....1 1 2 2....1 2 2 2....1 1 2 2
%e ...........................................
%e ...1 1 2 2....1 1 2 2....1 2 2 2...........
%e ...1 1 1 2....1 2 2 2....1 1 2 2...........
%Y A003168 (main diagonal), A071948, A193091, A211789 (row sums).
%K nonn,easy,tabl
%O 1,5
%A _Peter Bala_, Aug 02 2012
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