|
%I #14 Mar 03 2024 19:02:36
%S 5,22,105,567,3351,20676,129129,804817,4982759,30629206,187121865,
%T 1137631979,6891047527,41628865000,250987078681,1511105743781,
%U 9088662549303,54625229882746,328144877989145,1970524978549951
%N Number of nX4 binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.
%C Column 4 of A267245.
%H R. H. Hardin, <a href="/A267241/b267241.txt">Table of n, a(n) for n = 1..210</a>
%H Robert Israel, <a href="/A267241/a267241.pdf">Maple-assisted proof of empirical recurrence</a>
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (24, -246, 1420, -5121, 12084, -18944, 19536, -12720, 4736, -768).
%F Empirical: a(n) = 24*a(n-1) -246*a(n-2) +1420*a(n-3) -5121*a(n-4) +12084*a(n-5) -18944*a(n-6) +19536*a(n-7) -12720*a(n-8) +4736*a(n-9) -768*a(n-10).
%F Empirical formula verified (see link). - _Robert Israel_, Sep 08 2019
%e Some solutions for n=4
%e ..0..0..0..0....0..0..0..0....0..0..1..1....0..0..1..1....0..0..0..1
%e ..0..0..0..0....0..0..0..1....0..0..1..1....0..1..0..1....0..1..1..0
%e ..0..1..1..1....0..1..1..0....0..1..1..1....1..0..1..0....0..1..1..1
%e ..1..0..1..1....0..1..1..0....1..0..1..1....1..0..1..0....0..1..1..1
%p states:= select(proc(x) (x[1]=x[2] or x[5]=1) and (x[2]=x[3] or x[6]=1) and (x[3]=x[4] or x[7]=1) end proc, [seq(seq(seq(seq(seq(seq(seq([a,b,c,d,e,f,g],g=0..1),f=0..1),e=0..1),d=0..1),c=0..1),b=0..1),a=0..1)]):
%p T:= Matrix(54,54,proc(i,j) local k;
%p if add(states[j,k]-states[i,k],k=1..4) > 0 then return 0 fi;
%p if states[j,5]>states[i,5] or states[j,6]>states[i,6] or states[j,7]>states[i,7] then return 0 fi;
%p if states[i,1]>=states[i,2] and states[j,5]<> states[i,5] then return 0 fi;
%p if states[i,2]>=states[i,3] and states[j,6]<> states[i,6] then return 0 fi;
%p if states[i,3]>=states[i,4] and states[j,7]<> states[i,7] then return 0 fi;
%p 1
%p end proc):
%p U:= Vector(54,1):
%p E[0]:= Vector(54): E[0][1]:= 1:
%p for k from 1 to 25 do E[k]:= T . E[k-1] od:
%p seq(U^%T . E[j], j=1..25); # _Robert Israel_, Sep 08 2019
%t LinearRecurrence[{24, -246, 1420, -5121, 12084, -18944, 19536, -12720, 4736, -768}, {5, 22, 105, 567, 3351, 20676, 129129, 804817, 4982759, 30629206, 187121865}, 25] (* _Jean-François Alcover_, Oct 25 2022, after _Robert Israel_ *)
%Y Cf. A267245.
%K nonn
%O 1,1
%A _R. H. Hardin_, Jan 12 2016
|
