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Number of n X 4 nonnegative integer arrays with each row and column increasing from zero by 0 or 1.
1

%I #8 Jun 01 2018 11:57:38

%S 1,4,17,62,184,462,1022,2052,3819,6688,11143,17810,27482,41146,60012,

%T 85544,119493,163932,221293,294406,386540,501446,643402,817260,

%U 1028495,1283256,1588419,1951642,2381422,2887154,3479192,4168912,4968777,5892404

%N Number of n X 4 nonnegative integer arrays with each row and column increasing from zero by 0 or 1.

%C Column 4 of A202756.

%H R. H. Hardin, <a href="/A202752/b202752.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = (1/360)*n^6 + (1/30)*n^5 + (5/72)*n^4 - (1/6)*n^3 + (77/180)*n^2 + (19/30)*n.

%F Conjectures from _Colin Barker_, Jun 01 2018: (Start)

%F G.f.: x*(1 - 3*x + 10*x^2 - 8*x^3 + 2*x^4) / (1 - x)^7.

%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.

%F (End)

%e Some solutions for n=5:

%e ..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0

%e ..0..0..0..1....0..0..0..1....0..0..0..0....0..0..1..1....0..0..0..0

%e ..0..0..0..1....0..0..1..1....0..0..0..1....0..0..1..1....0..0..0..1

%e ..0..1..1..1....0..0..1..2....0..0..0..1....0..1..2..2....0..0..1..2

%e ..0..1..1..2....0..1..2..3....0..1..1..2....0..1..2..2....0..1..2..2

%Y Cf. A202756.

%K nonn

%O 1,2

%A _R. H. Hardin_, Dec 23 2011