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%I #12 Mar 17 2024 11:16:15
%S 7,50,475,6292,107015,2093467,43555569,924051709,19614050515,
%T 413556580944,8645774602327,179276181587698,3691120876565687,
%U 75550095426967737,1538986699132717645,31229753343696948035
%N Number of n X 6 binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.
%C Column 6 of A267245.
%H R. H. Hardin, <a href="/A267243/b267243.txt">Table of n, a(n) for n = 1..210</a>
%H Robert Israel, <a href="/A267243/a267243.pdf">Maple-assisted proof of empirical recurrence</a>
%H <a href="/index/Rec#order_22">Index entries for linear recurrences with constant coefficients</a>, signature (114, -5915, 186008, -3982785, 61835542, -723657627, 6549515604, -46652032035, 264676225246, -1205477853945, 4427867737616, -13139368875011, 31468929403866, -60602488003009, 93197329064964, -113220771193368, 106920682204032, -76630180181904, 40173465734208, -14497964755200, 3213273369600, -329204736000).
%F Empirical: a(n) = 114*a(n-1) - 5915*a(n-2) + 186008*a(n-3) - 3982785*a(n-4) + 61835542*a(n-5) - 723657627*a(n-6) + 6549515604*a(n-7) - 46652032035*a(n-8) + 264676225246*a(n-9) - 1205477853945*a(n-10) + 4427867737616*a(n-11) - 13139368875011*a(n-12) + 31468929403866*a(n-13) - 60602488003009*a(n-14) + 93197329064964*a(n-15) - 113220771193368*a(n-16) + 106920682204032*a(n-17) - 76630180181904*a(n-18) + 40173465734208*a(n-19) - 14497964755200*a(n-20) + 3213273369600*a(n-21) - 329204736000*a(n-22).
%F Empirical formula verified (see link). - _Robert Israel_, Sep 08 2019
%e Some solutions for n=4:
%e 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 1
%e 0 0 0 1 0 1 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 1
%e 0 1 1 1 1 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0
%e 0 0 1 1 1 1 0 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 1 1
%p S[2]:= [[0,0,0],[0,0,1],[0,1,1],[1,0,1],[1,1,0],[1,1,1]]:
%p for i from 3 to 6 do
%p S[i]:= map(proc(t) [op(t[1..i-1]),t[i-1],op(t[i..-1]),0], [op(t[1..i-1]),t[i-1],op(t[i..-1]),1],
%p [op(t[1..i-1]),1-t[i-1],op(t[i..-1]),1] end proc, S[i-1])
%p od:
%p states:= S[6]:
%p T:= Matrix(486,486,proc(i,j) local k;
%p if add(states[j,k]-states[i,k],k=1..6) > 0 then return 0 fi;
%p for k from 7 to 11 do if states[j,k]>states[i,k] then return 0 fi od;
%p for k from 1 to 5 do if states[i,k]>=states[i,k+1] and states[j,k+6]<>states[i,k+6] then return 0 fi od;
%p 1
%p end proc):
%p E:= Vector(486): E[1]:= 1:
%p U[0]:= Vector[row](486,1):
%p for k from 1 to 25 do U[k]:= U[k-1].T od:
%p seq(U[j] . E, j=1..25); # _Robert Israel_, Sep 08 2019
%Y Cf. A267245.
%K nonn
%O 1,1
%A _R. H. Hardin_, Jan 12 2016
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