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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

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

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