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

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

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

Table of n, a(n) for n=1..55.

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