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 A211788 Triangle enumerating certain two-line arrays of positive integers. 3
 1, 1, 1, 1, 4, 4, 1, 7, 21, 21, 1, 10, 47, 126, 126, 1, 13, 82, 324, 818, 818, 1, 16, 126, 642, 2300, 5594, 5594, 1, 19, 179, 1107, 4977, 16741, 39693, 39693, 1, 22, 241, 1746, 9335, 38642, 124383, 289510, 289510, 1, 25, 312, 2586, 15941, 77273, 301630, 939880, 2157150, 2157150 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS This is the table of f(n,k) in the notation of Carlitz (p.123). The triangle enumerates two-line arrays of positive integers ............a_1 a_2 ... a_n.......... ............b_1 b_2 ... b_n.......... such that 1) max(a_i, b_i) <= min(a_(i+1), b_(i+1)) for 1 <= i <= n-1 2) max(a_i, b_i) <= i for 1 <= i <= n 3) a_n = b_n = k. See A071948 and A193091 for other two-line array enumerations. LINKS L. Carlitz, Enumeration of two-line arrays, Fib. Quart., Vol. 11 Number 2 (1973), 113-130. FORMULA Recurrence equation: 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. 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)}. Row reverse has production matrix 1 1 3 3 1 5 5 3 1 7 7 5 3 1 ... Main diagonal T(n,n) = A003168(n). Row sums A211789. EXAMPLE Triangle begins .n\k.|..1....2....3....4....5....6 = = = = = = = = = = = = = = = = = = ..1..|..1 ..2..|..1....1 ..3..|..1....4....4 ..4..|..1....7...21...21 ..5..|..1...10...47..126..126 ..6..|..1...13...82..324..818..818 ... T(4,2) = 7: The 7 two-line arrays are ...1 1 1 2....1 1 2 2....1 2 2 2....1 1 1 2 ...1 1 1 2....1 1 2 2....1 2 2 2....1 1 2 2 ........................................... ...1 1 2 2....1 1 2 2....1 2 2 2........... ...1 1 1 2....1 2 2 2....1 1 2 2........... CROSSREFS A003168 (main diagonal), A071948, A193091, A211789 (row sums). Sequence in context: A047213 A128213 A171716 * A318732 A016706 A358204 Adjacent sequences: A211785 A211786 A211787 * A211789 A211790 A211791 KEYWORD nonn,easy,tabl AUTHOR Peter Bala, Aug 02 2012 STATUS approved

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Last modified November 27 21:40 EST 2022. Contains 358406 sequences. (Running on oeis4.)