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A211788 Triangle enumerating certain two-line arrays of positive integers.

%I A211788 #21 Jan 05 2025 19:51:39
%S A211788 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 A211788 642,2300,5594,5594,1,19,179,1107,4977,16741,39693,39693,1,22,241,
%U A211788 1746,9335,38642,124383,289510,289510,1,25,312,2586,15941,77273,301630,939880,2157150,2157150
%N A211788 Triangle enumerating certain two-line arrays of positive integers.
%C A211788 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 A211788 ............a_1 a_2 ... a_n..........
%C A211788 ............b_1 b_2 ... b_n..........
%C A211788 such that
%C A211788 1) max(a_i, b_i) <= min(a_(i+1), b_(i+1)) for 1 <= i <= n-1
%C A211788 2) max(a_i, b_i) <= i for 1 <= i <= n
%C A211788 3) a_n = b_n = k.
%C A211788 See A071948 and A193091 for other two-line array enumerations.
%C A211788 It appears that the row reverse array is the Riordan array (f(x), g(x)), where f(x) = 1 + x + 4*x^2 + 21*x^3 + 126*x^4 + 818*x^5 + ... is the g.f. of A003168 and g(x) = x + 3*x^2 + 14*x^3 + 79*x^4 + 494*x^5 + 3294*x^6 + ... is the g.f. of A003169. - _Peter Bala_, Nov 26 2024
%H A211788 L. Carlitz, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-2/carlitz.pdf">Enumeration of two-line arrays</a>, Fib. Quart., Vol. 11 Number 2 (1973), 113-130.
%F A211788 Recurrence equation:
%F A211788 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 A211788 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 A211788 Row reverse has production matrix
%F A211788   1 1
%F A211788   3 3 1
%F A211788   5 5 3 1
%F A211788   7 7 5 3 1
%F A211788   ...
%F A211788 Main diagonal T(n,n) = A003168(n). Row sums A211789.
%e A211788 Triangle begins
%e A211788 .n\k.|..1....2....3....4....5....6
%e A211788 = = = = = = = = = = = = = = = = = =
%e A211788 ..1..|..1
%e A211788 ..2..|..1....1
%e A211788 ..3..|..1....4....4
%e A211788 ..4..|..1....7...21...21
%e A211788 ..5..|..1...10...47..126..126
%e A211788 ..6..|..1...13...82..324..818..818
%e A211788 ...
%e A211788 T(4,2) = 7: The 7 two-line arrays are
%e A211788 ...1 1 1 2....1 1 2 2....1 2 2 2....1 1 1 2
%e A211788 ...1 1 1 2....1 1 2 2....1 2 2 2....1 1 2 2
%e A211788 ...........................................
%e A211788 ...1 1 2 2....1 1 2 2....1 2 2 2...........
%e A211788 ...1 1 1 2....1 2 2 2....1 1 2 2...........
%Y A211788 Cf. A003168 (main diagonal), A211789 (row sums).
%Y A211788 Cf. A003169, A071948, A193091.
%K A211788 nonn,easy,tabl
%O A211788 1,5
%A A211788 _Peter Bala_, Aug 02 2012