A200886 T(n,k) is the number of 0..k arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors.

7, 22, 12, 50, 51, 21, 95, 144, 121, 37, 161, 325, 422, 292, 65, 252, 636, 1121, 1268, 704, 114, 372, 1127, 2507, 3985, 3823, 1691, 200, 525, 1856, 4977, 10213, 14288, 11472, 4059, 351, 715, 2889, 9052, 22736, 42182, 50995, 34350, 9749, 616, 946, 4300, 15393

Offset: 1

Author

R. H. Hardin, Nov 23 2011

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of lattice points in k*P where P is a polytope of dimension n+2 in R^(n+2) whose vertices are lattice points, and therefore for each n it is an Ehrhart polynomial of degree n+2. This confirms the empirical formulas for the rows. - Robert Israel, Mar 21 2021

Examples

			Some solutions for n=4, k=3:
  1   2   3   0   0   1   2   3   0   1   2   3   3   1   2   2
  1   2   1   0   1   0   1   0   3   0   2   2   3   0   3   2
  2   2   3   0   2   2   3   2   3   0   3   3   3   1   3   0
  2   0   3   0   3   3   3   3   2   0   3   3   3   1   0   2
  1   1   2   1   3   3   2   3   0   1   3   3   3   1   2   3
  0   2   2   1   3   2   1   0   2   1   2   1   1   3   3   3
Table starts:
....7....22.....50......95......161.......252.......372........525........715
...12....51....144.....325......636......1127......1856.......2889.......4300
...21...121....422....1121.....2507......4977......9052......15393......24817
...37...292...1268....3985....10213.....22736.....45648......84681.....147565
...65...704...3823...14288....42182....105813....235538.....478467.....904111
..114..1691..11472...50995...173606....491533...1215616....2710413....5567530
..200..4059..34350..181336...710976...2269938...6233356...15250675...34054592
..351..9749.102896..644721..2908797..10462235..31868448...85473225..207289059
..616.23422.308419.2294193.11911516..48259083.163014678..479101189.1261310492
.1081.56268.924532.8166441.48807427.222798408.834763824.2688814689.7684922749
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..9999

Crossrefs

Column 1 is A005251(n+5).

Row 1 is A002412(n+1).

Formula

Empirical for columns:

k=1: a(n) = 2*a(n-1) -a(n-2) +a(n-3)

k=2: a(n) = 3*a(n-1) -3*a(n-2) +4*a(n-3) -a(n-4) +a(n-5)

k=3: a(n) = 4*a(n-1) -6*a(n-2) +10*a(n-3) -5*a(n-4) +6*a(n-5) -a(n-6) +a(n-7)

k=4: a(n) = 5*a(n-1) -10*a(n-2) +20*a(n-3) -15*a(n-4) +21*a(n-5) -7*a(n-6) +8*a(n-7) -a(n-8) +a(n-9)

k=5: a(n) = 6*a(n-1) -15*a(n-2) +35*a(n-3) -35*a(n-4) +56*a(n-5) -28*a(n-6) +36*a(n-7) -9*a(n-8) +10*a(n-9) -a(n-10) +a(n-11)

k=6: a(n) = 7*a(n-1) -21*a(n-2) +56*a(n-3) -70*a(n-4) +126*a(n-5) -84*a(n-6) +120*a(n-7) -45*a(n-8) +55*a(n-9) -11*a(n-10) +12*a(n-11) -a(n-12) +a(n-13)

k=7: a(n) = 8*a(n-1) -28*a(n-2) +84*a(n-3) -126*a(n-4) +252*a(n-5) -210*a(n-6) +330*a(n-7) -165*a(n-8) +220*a(n-9) -66*a(n-10) +78*a(n-11) -13*a(n-12) +14*a(n-13) -a(n-14) +a(n-15)

Empirical for rows:

n=1: a(k) = (2/3)*k^3 + (5/2)*k^2 + (17/6)*k + 1

n=2: a(k) = (1/3)*k^4 + (7/3)*k^3 + (14/3)*k^2 + (11/3)*k + 1

n=3: a(k) = (2/15)*k^5 + (11/6)*k^4 + (35/6)*k^3 + (23/3)*k^2 + (68/15)*k + 1

n=4: a(k) = (2/45)*k^6 + (19/15)*k^5 + (217/36)*k^4 + (71/6)*k^3 + (2057/180)*k^2 + (27/5)*k + 1

n=5: a(k) = (4/315)*k^7 + (7/9)*k^6 + (241/45)*k^5 + (1067/72)*k^4 + (3757/180)*k^3 + (1145/72)*k^2 + (2629/420)*k + 1

n=6: a(k) = (1/315)*k^8 + (134/315)*k^7 + (21/5)*k^6 + (571/36)*k^5 + (1841/60)*k^4 + (6047/180)*k^3 + (26603/1260)*k^2 + (299/42)*k + 1

n=7: a(k) = (2/2835)*k^9 + (131/630)*k^8 + (2803/945)*k^7 + (1349/90)*k^6 + (41449/1080)*k^5 + (20423/360)*k^4 + (1149293/22680)*k^3 + (22741/840)*k^2 + (2011/252)*k + 1