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A267244 Number of n X 7 binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

%I A267244 #12 Mar 17 2024 11:17:18
%S A267244 8,70,904,18205,516084,17892539,683027146,27044976947,1079112886476,
%T A267244 42860145907558,1687239907979286,65777529883058423,
%U A267244 2540922972496976428,97351678797063744735,3703224984260808730288,139993814565092144904305
%N A267244 Number of n X 7 binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.
%C A267244 Column 7 of A267245.
%H A267244 R. H. Hardin, <a href="/A267244/b267244.txt">Table of n, a(n) for n = 1..210</a>
%H A267244 Robert Israel, <a href="/A267244/a267244.pdf">Maple-assisted proof of empirical formula</a>
%H A267244 <a href="/index/Rec#order_32">Index entries for linear recurrences with constant coefficients</a>, signature (236, -25680, 1717504, -79417394, 2707798440, -70899406188, 1465896913824, -24421757248431, 332861244138564, -3755300016546300, 35390628699049728, -280610566308801516, 1882413463252467120, -10729331312513919192, 52123544280277991616, -216277785263000273775, 767370439659990868020, -2328674591889971376488, 6039623808173911907968, -13364805995823788545362, 25161918805489259088488, -40140151907739595227388, 53955000634729356546720, -60650423670523051920321, 56445409553303282568732, -42910761548685014780364, 26160176586524646234240, -12460398044348337274800, 4460624272170497592000, -1127355192728480520000, 179146175950526400000, -13449500030736000000).
%F A267244 Empirical: a(n) = 236*a(n-1) - 25680*a(n-2) + 1717504*a(n-3) - 79417394*a(n-4) + 2707798440*a(n-5) - 70899406188*a(n-6) + 1465896913824*a(n-7) - 24421757248431*a(n-8) + 332861244138564*a(n-9) - 3755300016546300*a(n-10) + 35390628699049728*a(n-11) - 280610566308801516*a(n-12) + 1882413463252467120*a(n-13) - 10729331312513919192*a(n-14) + 52123544280277991616*a(n-15) - 216277785263000273775*a(n-16) + 767370439659990868020*a(n-17) - 2328674591889971376488*a(n-18) + 6039623808173911907968*a(n-19) - 13364805995823788545362*a(n-20) + 25161918805489259088488*a(n-21) - 40140151907739595227388*a(n-22) + 53955000634729356546720*a(n-23) - 60650423670523051920321*a(n-24) + 56445409553303282568732*a(n-25) - 42910761548685014780364*a(n-26) + 26160176586524646234240*a(n-27) - 12460398044348337274800*a(n-28) + 4460624272170497592000*a(n-29) - 1127355192728480520000*a(n-30) + 179146175950526400000*a(n-31) - 13449500030736000000*a(n-32).
%F A267244 Empirical formula verified (see link). - _Robert Israel_, Sep 08 2019
%e A267244 Some solutions for n=4:
%e A267244   0 0 0 0 0 0 0    0 0 0 0 0 1 1    0 0 0 0 0 0 1
%e A267244   0 0 0 0 0 0 0    0 0 0 0 1 0 1    0 0 0 1 1 1 0
%e A267244   0 0 0 0 1 1 1    0 0 1 1 0 0 0    0 0 1 0 1 1 0
%e A267244   0 0 1 1 0 1 1    0 1 1 1 1 0 0    0 0 1 1 0 1 0
%p A267244 S[2]:= [[0,0,0],[0,0,1],[0,1,1],[1,0,1],[1,1,0],[1,1,1]]:
%p A267244 for i from 3 to 7 do
%p A267244   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 A267244      [op(t[1..i-1]),1-t[i-1],op(t[i..-1]),1] end proc, S[i-1])
%p A267244 od:
%p A267244 states:= S[7]:
%p A267244 T:= Matrix(1458,1458,proc(i,j) local k;
%p A267244   if add(states[j,k]-states[i,k],k=1..7) > 0 then return 0 fi;
%p A267244   for k from 8 to 13 do if states[j,k]>states[i,k] then return 0 fi od;
%p A267244   for k from 1 to 6 do if states[i,k]>=states[i,k+1] and states[j,k+7]<>states[i,k+7] then return 0 fi od;
%p A267244 1
%p A267244 end proc):
%p A267244 E:= Vector(1458): E[1]:= 1:
%p A267244 U[0]:= Vector[row](1458,1):
%p A267244 for k from 1 to 32 do U[k]:= U[k-1].T od:
%p A267244 seq(U[j] . E, j=1..32);  # _Robert Israel_, Sep 08 2019
%Y A267244 Cf. A267245.
%K A267244 nonn
%O A267244 1,1
%A A267244 _R. H. Hardin_, Jan 12 2016