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Number of n X 5 0..1 arrays with rows unimodal and columns nondecreasing.
3

%I #41 Feb 22 2024 20:31:34

%S 1,16,86,296,791,1792,3612,6672,11517,18832,29458,44408,64883,92288,

%T 128248,174624,233529,307344,398734,510664,646415,809600,1004180,

%U 1234480,1505205,1821456,2188746,2613016,3100651,3658496,4293872,5014592,5828977

%N Number of n X 5 0..1 arrays with rows unimodal and columns nondecreasing.

%H Andrew Howroyd, <a href="/A225007/b225007.txt">Table of n, a(n) for n = 0..1000</a> (terms 1..210 from R. H. Hardin)

%F a(n) = (2/15)*n^5 + (7/6)*n^4 + (23/6)*n^3 + (35/6)*n^2 + (121/30)*n + 1.

%F 6*a(n) = Sum_{i=1..n+1} A000384(i)*A000384(i+1). - _Bruno Berselli_, Feb 05 2014

%F From _Colin Barker_, Mar 16 2018: (Start)

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n >= 6.

%F G.f.: (1 + 10*x + 5*x^2) / (1 - x)^6. (End)

%e Some solutions for n=3:

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

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

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

%e 6*a(7) = 40032 = 1*6 + 6*15 + 15*28 + 28*45 + 45*66 + 66*91 + 91*120 + 120*153. - _Bruno Berselli_, Feb 05 2014

%Y Column 5 of A225010.

%K nonn,easy

%O 0,2

%A _R. H. Hardin_, Apr 23 2013

%E a(0)=1 prepended by _Andrew Howroyd_, Feb 11 2024