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 A267242 Number of nX5 binary arrays with row sums nondecreasing and columns lexicographically nondecreasing. 1

%I #10 Mar 17 2024 11:15:02

%S 6,34,232,1986,20040,220235,2499080,28501471,323067002,3626695952,

%T 40306404192,443852375808,4848323701804,52590398731297,

%U 567018802063680,6081537709403509,64929807220896558,690446673537426382

%N Number of nX5 binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

%C Column 5 of A267245.

%H R. H. Hardin, <a href="/A267242/b267242.txt">Table of n, a(n) for n = 1..210</a>

%H Robert Israel, <a href="/A267242/a267242.pdf">Maple-assisted proof of empirical formula</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (52, -1196, 16140, -142918, 879116, -3875668, 12442580, -29232481, 50015232, -61355336, 52355680, -29405200, 9744000, -1440000).

%F Empirical: a(n) = 52*a(n-1) -1196*a(n-2) +16140*a(n-3) -142918*a(n-4) +879116*a(n-5) -3875668*a(n-6) +12442580*a(n-7) -29232481*a(n-8) +50015232*a(n-9) -61355336*a(n-10) +52355680*a(n-11) -29405200*a(n-12) +9744000*a(n-13) -1440000*a(n-14).

%F Empirical formula verified (see link). - _Robert Israel_, Sep 08 2019

%e Some solutions for n=4

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

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

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

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

%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 5 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[5]:

%p T:= Matrix(162,162,proc(i,j) local k;

%p if add(states[j,k]-states[i,k],k=1..5) > 0 then return 0 fi;

%p for k from 6 to 9 do if states[j,k]>states[i,k] then return 0 fi od;

%p for k from 1 to 4 do if states[i,k]>=states[i,k+1] and states[j,k+5]<>states[i,k+5] then return 0 fi od;

%p 1

%p end proc):

%p E:= Vector(162): E[1]:= 1:

%p U[0]:= Vector[row](162,1):

%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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Last modified April 18 22:18 EDT 2024. Contains 371782 sequences. (Running on oeis4.)