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A200871
T(n,k)=Number of 0..k arrays x(0..n+1) of n+2 elements without any interior element greater than both neighbors or less than both neighbors
13
6, 17, 10, 36, 37, 16, 65, 94, 77, 26, 106, 195, 236, 163, 42, 161, 356, 567, 602, 343, 68, 232, 595, 1168, 1673, 1528, 723, 110, 321, 932, 2163, 3886, 4917, 3882, 1523, 178, 430, 1389, 3704, 7973, 12890, 14455, 9858, 3209, 288, 561, 1990, 5973, 14932, 29325
OFFSET
1,1
COMMENTS
Table starts
...6....17.....36......65.....106......161......232.......321.......430
..10....37.....94.....195.....356......595......932......1389......1990
..16....77....236.....567....1168.....2163.....3704......5973......9184
..26...163....602....1673....3886.....7973....14932.....26073.....43066
..42...343...1528....4917...12890....29325....60112....113745....201994
..68...723...3882...14455...42744...107777...241718....495495....945790
.110..1523...9858...42479..141688...395929...971416...2156867...4424298
.178..3209..25038..124851..469726..1454643..3904290...9389377..20696974
.288..6761..63592..366959.1557320..5344795.15693816..40880321..96838448
.466.14245.161514.1078565.5163158.19638715.63085186.177996275.453123270
LINKS
FORMULA
Empirical for columns:
k=1: a(n) = a(n-1) +a(n-2)
k=2: a(n) = 2*a(n-1) +a(n-4)
k=3: a(n) = 2*a(n-1) +a(n-2) +2*a(n-4) +a(n-5)
k=4: a(n) = 3*a(n-1) -a(n-2) +a(n-3) +4*a(n-4) +a(n-6) +a(n-7)
k=5: a(n) = 3*a(n-1) +a(n-3) +7*a(n-4) +3*a(n-5) +2*a(n-6) +3*a(n-7) +a(n-8)
k=6: a(n) = 4*a(n-1) -3*a(n-2) +4*a(n-3) +9*a(n-4) +7*a(n-6) +6*a(n-7) +a(n-8) +2*a(n-9) +a(n-10)
k=7: a(n) = 4*a(n-1) -2*a(n-2) +4*a(n-3) +15*a(n-4) +6*a(n-5) +12*a(n-6) +16*a(n-7) +7*a(n-8) +5*a(n-9) +4*a(n-10) +a(n-11)
Empirical for rows:
n=1: a(k) = (1/3)*k^3 + 2*k^2 + (8/3)*k + 1
n=2: a(k) = (1/12)*k^4 + (3/2)*k^3 + (47/12)*k^2 + (7/2)*k + 1
n=3: a(k) = (1/60)*k^5 + (3/4)*k^4 + (15/4)*k^3 + (25/4)*k^2 + (127/30)*k + 1
n=4: a(k) = (1/360)*k^6 + (7/24)*k^5 + (197/72)*k^4 + (185/24)*k^3 + (1667/180)*k^2 + 5*k + 1
n=5: a(k) = (1/2520)*k^7 + (17/180)*k^6 + (281/180)*k^5 + (64/9)*k^4 + (4927/360)*k^3 + (2303/180)*k^2 + (604/105)*k + 1
n=6: a(k) = (1/20160)*k^8 + (19/720)*k^7 + (211/288)*k^6 + (1889/360)*k^5 + (44167/2880)*k^4 + (15991/720)*k^3 + (5689/336)*k^2 + (391/60)*k + 1
n=7: a(k) = (1/181440)*k^9 + (131/20160)*k^8 + (8893/30240)*k^7 + (4621/1440)*k^6 + (118933/8640)*k^5 + (83957/2880)*k^4 + (763489/22680)*k^3 + (36343/1680)*k^2 + (9169/1260)*k + 1
EXAMPLE
Some solutions for n=4 k=3
..3....2....0....0....2....0....1....0....0....2....3....3....1....1....1....3
..2....2....0....2....0....2....2....2....0....3....1....3....2....1....2....3
..2....1....3....3....0....2....2....2....0....3....0....3....2....2....2....3
..2....0....3....3....3....0....1....0....2....3....0....2....2....2....0....2
..2....0....0....1....3....0....1....0....2....3....0....2....2....2....0....2
..3....2....0....1....1....0....2....3....3....2....2....1....3....2....0....0
MATHEMATICA
t[0, k_, x_, y_] := 1; t[n_, k_, x_, y_] := t[n, k, x, y] = Sum[If[z <= x <= y || y <= x <= z, t[n-1, k, z, x], 0], {z, k+1}]; t[n_, k_] := Sum[t[n, k, x, y], {x, k+1}, {y, k+1}]; TableForm@ Table[t[n, k], {n, 8}, {k, 8}] (* Giovanni Resta, Mar 05 2014 *)
CROSSREFS
Column 1 is A006355(n+4)
Row 1 is A084990(n+1)
Sequence in context: A323516 A120930 A070395 * A112366 A095421 A063584
KEYWORD
nonn,tabl
AUTHOR
R. H. Hardin Nov 23 2011
STATUS
approved