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

1,1

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)

Links

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

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

Example

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

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 *)

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

Sequence in context: A361001 A223838 A224146 * A235494 A292960 A292963

Adjacent sequences: A225007 A225008 A225009 * A225011 A225012 A225013

Keyword

nonn,tabl

Author

R. H. Hardin, Apr 23 2013

Status

approved