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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. 13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

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

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

EXAMPLE

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

CROSSREFS

Column 1 is A005251(n+5).

Row 1 is A002412(n+1).

Sequence in context: A130740 A101119 A217014 * A070412 A286572 A055575

Adjacent sequences:  A200883 A200884 A200885 * A200887 A200888 A200889

KEYWORD

nonn,tabl

AUTHOR

R. H. Hardin, Nov 23 2011

STATUS

approved

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Last modified May 18 18:42 EDT 2022. Contains 353824 sequences. (Running on oeis4.)