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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.

%I A200871 #12 Jul 22 2025 15:59:16
%S A200871 6,17,10,36,37,16,65,94,77,26,106,195,236,163,42,161,356,567,602,343,
%T A200871 68,232,595,1168,1673,1528,723,110,321,932,2163,3886,4917,3882,1523,
%U A200871 178,430,1389,3704,7973,12890,14455,9858,3209,288,561,1990,5973,14932,29325
%N 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.
%C A200871 Table starts
%C A200871 ...6....17.....36......65.....106......161......232.......321.......430
%C A200871 ..10....37.....94.....195.....356......595......932......1389......1990
%C A200871 ..16....77....236.....567....1168.....2163.....3704......5973......9184
%C A200871 ..26...163....602....1673....3886.....7973....14932.....26073.....43066
%C A200871 ..42...343...1528....4917...12890....29325....60112....113745....201994
%C A200871 ..68...723...3882...14455...42744...107777...241718....495495....945790
%C A200871 .110..1523...9858...42479..141688...395929...971416...2156867...4424298
%C A200871 .178..3209..25038..124851..469726..1454643..3904290...9389377..20696974
%C A200871 .288..6761..63592..366959.1557320..5344795.15693816..40880321..96838448
%C A200871 .466.14245.161514.1078565.5163158.19638715.63085186.177996275.453123270
%H A200871 R. H. Hardin, <a href="/A200871/b200871.txt">Table of n, a(n) for n = 1..9999</a>
%F A200871 Empirical for columns:
%F A200871 k=1: a(n) = a(n-1) +a(n-2)
%F A200871 k=2: a(n) = 2*a(n-1) +a(n-4)
%F A200871 k=3: a(n) = 2*a(n-1) +a(n-2) +2*a(n-4) +a(n-5)
%F A200871 k=4: a(n) = 3*a(n-1) -a(n-2) +a(n-3) +4*a(n-4) +a(n-6) +a(n-7)
%F A200871 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)
%F A200871 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)
%F A200871 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)
%F A200871 Empirical for rows:
%F A200871 n=1: a(k) = (1/3)*k^3 + 2*k^2 + (8/3)*k + 1
%F A200871 n=2: a(k) = (1/12)*k^4 + (3/2)*k^3 + (47/12)*k^2 + (7/2)*k + 1
%F A200871 n=3: a(k) = (1/60)*k^5 + (3/4)*k^4 + (15/4)*k^3 + (25/4)*k^2 + (127/30)*k + 1
%F A200871 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
%F A200871 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
%F A200871 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
%F A200871 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
%e A200871 Some solutions for n=4 k=3
%e A200871 ..3....2....0....0....2....0....1....0....0....2....3....3....1....1....1....3
%e A200871 ..2....2....0....2....0....2....2....2....0....3....1....3....2....1....2....3
%e A200871 ..2....1....3....3....0....2....2....2....0....3....0....3....2....2....2....3
%e A200871 ..2....0....3....3....3....0....1....0....2....3....0....2....2....2....0....2
%e A200871 ..2....0....0....1....3....0....1....0....2....3....0....2....2....2....0....2
%e A200871 ..3....2....0....1....1....0....2....3....3....2....2....1....3....2....0....0
%t A200871 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 *)
%Y A200871 Column 1 is A006355(n+4)
%Y A200871 Row 1 is A084990(n+1)
%K A200871 nonn,tabl
%O A200871 1,1
%A A200871 _R. H. Hardin_ Nov 23 2011