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

8, 70, 904, 18205, 516084, 17892539, 683027146, 27044976947, 1079112886476, 42860145907558, 1687239907979286, 65777529883058423, 2540922972496976428, 97351678797063744735, 3703224984260808730288, 139993814565092144904305

Offset

1,1

Comments

Column 7 of A267245.

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Robert Israel, Maple-assisted proof of empirical formula

Index entries for linear recurrences with constant coefficients, 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).

Formula

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).

Empirical formula verified (see link). - Robert Israel, Sep 08 2019

Example

Some solutions for n=4:
  0 0 0 0 0 0 0    0 0 0 0 0 1 1    0 0 0 0 0 0 1
  0 0 0 0 0 0 0    0 0 0 0 1 0 1    0 0 0 1 1 1 0
  0 0 0 0 1 1 1    0 0 1 1 0 0 0    0 0 1 0 1 1 0
  0 0 1 1 0 1 1    0 1 1 1 1 0 0    0 0 1 1 0 1 0

Maple

S[2]:= [[0, 0, 0], [0, 0, 1], [0, 1, 1], [1, 0, 1], [1, 1, 0], [1, 1, 1]]:
for i from 3 to 7 do
  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],
     [op(t[1..i-1]), 1-t[i-1], op(t[i..-1]), 1] end proc, S[i-1])
od:
states:= S[7]:
T:= Matrix(1458, 1458, proc(i, j) local k;
  if add(states[j, k]-states[i, k], k=1..7) > 0 then return 0 fi;
  for k from 8 to 13 do if states[j, k]>states[i, k] then return 0 fi od;
  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;
1
end proc):
E:= Vector(1458): E[1]:= 1:
U[0]:= Vector[row](1458, 1):
for k from 1 to 32 do U[k]:= U[k-1].T od:
seq(U[j] . E, j=1..32);  # Robert Israel, Sep 08 2019

Crossrefs

Cf. A267245.

Sequence in context: A287482 A053729 A266433 * A228388 A020532 A043086

Adjacent sequences: A267241 A267242 A267243 * A267245 A267246 A267247

Keyword

nonn

Author

R. H. Hardin, Jan 12 2016

Status

approved