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A221967 T(n,k)=Number of -k..k arrays of length n with the sum ahead of each element differing from the sum following that element by k or less.

%I A221967 #6 Jul 23 2025 02:22:06
%S A221967 3,5,9,7,25,15,9,49,65,33,11,81,175,225,63,13,121,369,833,705,129,15,
%T A221967 169,671,2241,3647,2305,255,17,225,1105,4961,12609,16513,7425,513,19,
%U A221967 289,1695,9633,34111,73089,73983,24065,1023,21,361,2465,17025,78273,241153
%N A221967 T(n,k)=Number of -k..k arrays of length n with the sum ahead of each element differing from the sum following that element by k or less.
%C A221967 Table starts
%C A221967 ....3.......5.........7..........9..........11...........13............15
%C A221967 ....9......25........49.........81.........121..........169...........225
%C A221967 ...15......65.......175........369.........671.........1105..........1695
%C A221967 ...33.....225.......833.......2241........4961.........9633.........17025
%C A221967 ...63.....705......3647......12609.......34111........78273........159615
%C A221967 ..129....2305.....16513......73089......241153.......653185.......1535745
%C A221967 ..255....7425.....73983.....419841.....1690623......5407233......14661375
%C A221967 ..513...24065....332801....2419713....11888129.....44890625.....140355585
%C A221967 .1023...77825...1495039...13930497....83512319....372332545....1342437375
%C A221967 .2049..251905...6719489...80230401...586864641...3089205249...12843782145
%C A221967 .4095..815105..30195711..462012417..4123582463..25628045313..122870296575
%C A221967 .8193.2637825.135700481.2660655105.28975366145.212618141697.1175482548225
%H A221967 R. H. Hardin, <a href="/A221967/b221967.txt">Table of n, a(n) for n = 1..334</a>
%F A221967 Empirical for column k:
%F A221967 k=1: a(n) = a(n-1) +2*a(n-2)
%F A221967 k=2: a(n) = 3*a(n-1) +2*a(n-2) -4*a(n-3)
%F A221967 k=3: a(n) = 3*a(n-1) +8*a(n-2) -4*a(n-3) -8*a(n-4)
%F A221967 k=4: a(n) = 5*a(n-1) +8*a(n-2) -20*a(n-3) -8*a(n-4) +16*a(n-5)
%F A221967 k=5: a(n) = 5*a(n-1) +18*a(n-2) -20*a(n-3) -48*a(n-4) +16*a(n-5) +32*a(n-6)
%F A221967 k=6: a(n) = 7*a(n-1) +18*a(n-2) -56*a(n-3) -48*a(n-4) +112*a(n-5) +32*a(n-6) -64*a(n-7)
%F A221967 k=7: a(n) = 7*a(n-1) +32*a(n-2) -56*a(n-3) -160*a(n-4) +112*a(n-5) +256*a(n-6) -64*a(n-7) -128*a(n-8)
%F A221967 Empirical for row n:
%F A221967 n=1: a(n) = 2*n + 1
%F A221967 n=2: a(n) = 4*n^2 + 4*n + 1
%F A221967 n=3: a(n) = 4*n^3 + 6*n^2 + 4*n + 1
%F A221967 n=4: a(n) = (16/3)*n^4 + (32/3)*n^3 + (32/3)*n^2 + (16/3)*n + 1
%F A221967 n=5: a(n) = (20/3)*n^5 + (50/3)*n^4 + 20*n^3 + (40/3)*n^2 + (16/3)*n + 1
%F A221967 n=6: a(n) = (128/15)*n^6 + (128/5)*n^5 + (112/3)*n^4 + 32*n^3 + (272/15)*n^2 + (32/5)*n + 1
%F A221967 n=7: a(n) = (488/45)*n^7 + (1708/45)*n^6 + (2912/45)*n^5 + (602/9)*n^4 + (2072/45)*n^3 + (952/45)*n^2 + (32/5)*n + 1
%e A221967 Some solutions for n=6 k=4
%e A221967 ..4...-2....4....1...-4...-1...-2....1...-2...-1....1....3....4....1...-1...-1
%e A221967 .-4....4...-4....0....4....4....3....2....3....2...-2...-4...-2...-3....3....3
%e A221967 ..1...-3....3...-2...-1...-2...-3...-2...-2....2....0....3....1....2....0...-1
%e A221967 ..0....2...-1....3...-2....0....2....2....3...-3....4....1...-3...-2...-2....1
%e A221967 ..3...-4...-2...-3....3....3...-2....1...-1....0...-1...-3....0....3...-3...-2
%e A221967 ..1....1....2....1...-1...-2...-1....1...-1....1....0....1....2...-4....4....2
%Y A221967 Column 1 is A062510(n+1)
%Y A221967 Column 2 is A189318
%Y A221967 Row 2 is A016754
%Y A221967 Row 3 is A005917(n+1)
%Y A221967 Row 4 is A142993
%K A221967 nonn,tabl
%O A221967 1,1
%A A221967 _R. H. Hardin_ Feb 01 2013