A225010 T(n,k) = number of n X k 0..1 arrays with rows unimodal and columns nondecreasing.

2, 4, 3, 7, 9, 4, 11, 22, 16, 5, 16, 46, 50, 25, 6, 22, 86, 130, 95, 36, 7, 29, 148, 296, 295, 161, 49, 8, 37, 239, 610, 791, 581, 252, 64, 9, 46, 367, 1163, 1897, 1792, 1036, 372, 81, 10, 56, 541, 2083, 4166, 4900, 3612, 1716, 525, 100, 11, 67, 771, 3544, 8518, 12174, 11088, 6672, 2685, 715, 121, 12

Offset: 1

Author

R. H. Hardin, Apr 23 2013

Keywords

• nonn
• tabl

Comments

Table starts

..2...4...7...11....16.....22.....29......37......46.......56.......67

..3...9..22...46....86....148....239.....367.....541......771.....1068

..4..16..50..130...296....610...1163....2083....3544.....5776.....9076

..5..25..95..295...791...1897...4166....8518...16414....30086....52834

..6..36.161..581..1792...4900..12174...27966...60172...122464...237590

..7..49.252.1036..3612..11088..30738...78354..186142...416394...884236

..8..64.372.1716..6672..22716..69498..194634..505912..1233584..2845492

..9..81.525.2685.11517..43065.144111..439791.1241383..3276559..8157227

.10.100.715.4015.18832..76714.278707..920491.2803658..7963384.21280337

.11.121.946.5786.29458.129844.508937.1808521.5911763.17978389.51325352

From Charles A. Lane, Aug 22 2013: (Start)

The first column is also the coefficients of a in y''[x] - a*x^n*y[x] + b*en*y[x] = 0 where n = 0. The recursion yields coefficients of a, a*b*en, a*b^2*en^2 etc.

The second column is obtained when n=1, the third column when n=2. The final column is for n=10.

Example: Write a normal recursion for n=4. For convenience set x to 1. Running the recursion yields

1-(b en)/2+(b^2 en^2)/24+1/30 (a-(b^3 en^3)/24)+(-384 a b en+b^4 en^4)/40320+(2064 a b^2 en^2-b^5 en^5)/3628800+(120960 a^2-7104 a b^3 en^3+b^6 en^6)/479001600+(-4682880 a^2 b en+18984 a b^4 en^4-b^7 en^7)/87178291200+(54268416 a^2 b^2 en^2-43008 a b^5 en^5+b^8 en^8)/20922789888000.

The coefficient of a is 24, the coefficient of a b en is 384 and the coefficient of a b^2 en^2 is 2064. Dividing by 4! yields a sequence of 1,16,86... , the same as column 5 without the leading 1. There is a hint of unity among the oscillators. (End)

Examples

			Some solutions for n=3 k=4
..0..0..0..0....0..1..0..0....0..0..0..0....1..1..1..1....0..0..0..0
..0..0..0..0....0..1..1..0....0..0..0..0....1..1..1..1....1..1..0..0
..0..0..0..1....1..1..1..0....1..1..0..0....1..1..1..1....1..1..1..1
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..5304

Crossrefs

Column 2 is A000290(n+1).

Column 3 is A002412(n+1).

Column 4 is A006324(n+1).

Row 1 is A000124.

Row 2 is A223718.

Row 3 is A223659.

Cf. A071920, A071921 (larger and reflected versions of table). - Alois P. Heinz, Sep 22 2013

Programs

Maple

T:= (n, k)-> add(binomial(k+2*j-1, 2*j), j=0..n):
seq(seq(T(n, 1+d-n), n=1..d), d=1..12);  # Alois P. Heinz, Sep 22 2013

Mathematica

T[n_, k_] := Sum[Binomial[k + 2*j - 1, 2*j], {j, 0, n}]; Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Apr 07 2016, after Alois P. Heinz *)

Formula

Empirical: columns k=1..7 are polynomials of degree k.

Empirical: rows n=1..7 are polynomials of degree 2n.

T(n,k) = Sum_{j=0..n} C(k+2*j-1,2*j). - Alois P. Heinz, Sep 22 2013