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%I #15 Mar 22 2021 06:27:37
%S 7,22,12,50,51,21,95,144,121,37,161,325,422,292,65,252,636,1121,1268,
%T 704,114,372,1127,2507,3985,3823,1691,200,525,1856,4977,10213,14288,
%U 11472,4059,351,715,2889,9052,22736,42182,50995,34350,9749,616,946,4300,15393
%N 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.
%C 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
%H R. H. Hardin, <a href="/A200886/b200886.txt">Table of n, a(n) for n = 1..9999</a>
%F Empirical for columns:
%F k=1: a(n) = 2*a(n-1) -a(n-2) +a(n-3)
%F k=2: a(n) = 3*a(n-1) -3*a(n-2) +4*a(n-3) -a(n-4) +a(n-5)
%F 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)
%F 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)
%F 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)
%F 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)
%F 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)
%F Empirical for rows:
%F n=1: a(k) = (2/3)*k^3 + (5/2)*k^2 + (17/6)*k + 1
%F n=2: a(k) = (1/3)*k^4 + (7/3)*k^3 + (14/3)*k^2 + (11/3)*k + 1
%F n=3: a(k) = (2/15)*k^5 + (11/6)*k^4 + (35/6)*k^3 + (23/3)*k^2 + (68/15)*k + 1
%F 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
%F 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
%F 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
%F 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
%e Some solutions for n=4, k=3:
%e 1 2 3 0 0 1 2 3 0 1 2 3 3 1 2 2
%e 1 2 1 0 1 0 1 0 3 0 2 2 3 0 3 2
%e 2 2 3 0 2 2 3 2 3 0 3 3 3 1 3 0
%e 2 0 3 0 3 3 3 3 2 0 3 3 3 1 0 2
%e 1 1 2 1 3 3 2 3 0 1 3 3 3 1 2 3
%e 0 2 2 1 3 2 1 0 2 1 2 1 1 3 3 3
%e Table starts:
%e ....7....22.....50......95......161.......252.......372........525........715
%e ...12....51....144.....325......636......1127......1856.......2889.......4300
%e ...21...121....422....1121.....2507......4977......9052......15393......24817
%e ...37...292...1268....3985....10213.....22736.....45648......84681.....147565
%e ...65...704...3823...14288....42182....105813....235538.....478467.....904111
%e ..114..1691..11472...50995...173606....491533...1215616....2710413....5567530
%e ..200..4059..34350..181336...710976...2269938...6233356...15250675...34054592
%e ..351..9749.102896..644721..2908797..10462235..31868448...85473225..207289059
%e ..616.23422.308419.2294193.11911516..48259083.163014678..479101189.1261310492
%e .1081.56268.924532.8166441.48807427.222798408.834763824.2688814689.7684922749
%Y Column 1 is A005251(n+5).
%Y Row 1 is A002412(n+1).
%K nonn,tabl
%O 1,1
%A _R. H. Hardin_, Nov 23 2011
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