A267245 -id:A267245 - cp's OEIS frontend

A233302 Number of (2+1) X (n+1) 0..1 arrays x(i,j) with row sums sum{x(i,j), j=1..n+1} nondecreasing, and column sums sum{i^2*x(i,j), i=1..2+1} nondecreasing.

15, 42, 105, 232, 475, 904, 1632, 2806, 4642, 7414, 11500, 17368, 25636, 37054, 52579, 73354, 100799, 136586, 182749, 241654, 316129, 409430, 525392, 668390, 843514, 1056524, 1314050, 1623542, 1993496, 2433398, 2953979, 3567152, 4286297, 5126192

Offset: 1

R. H. Hardin, Dec 07 2013

nonn

Row 2 of A233301, row 3 of A267245.

			Some solutions for n=5:
..0..0..0..0..1..0....1..1..0..0..0..0....0..0..1..0..0..0....0..0..0..0..0..1
..0..0..0..0..0..1....0..0..1..1..1..1....0..0..0..0..0..1....0..1..1..1..1..1
..0..0..0..1..1..1....0..1..1..1..1..1....0..0..0..1..1..1....0..1..1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..172

Cf. A233301, A267245.

Empirical: a(n) = 4*a(n-1) - 3*a(n-2) - 7*a(n-3) + 10*a(n-4) + 3*a(n-5) - 6*a(n-6) - 6*a(n-7) + 3*a(n-8) + 10*a(n-9) - 7*a(n-10) - 3*a(n-11) + 4*a(n-12) - a(n-13). - R. H. Hardin, Jan 17 2016.

Empirical g.f.: x*(15 - 18*x - 18*x^2 + 43*x^3 + 6*x^4 - 30*x^5 - 21*x^6 + 22*x^7 + 33*x^8 - 31*x^9 - 8*x^10 + 15*x^11 - 4*x^12) / ((1 - x)^8*(1 + x)^3*(1 + x + x^2)). - Colin Barker, Mar 19 2018

A233303 Number of (3+1)X(n+1) 0..1 arrays x(i,j) with row sums sum{x(i,j), j=1..n+1} nondecreasing, and column sums sum{i^2*x(i,j), i=1..3+1} nondecreasing.

Original entry on oeis.org

31, 141, 567, 1986, 6292, 18205, 48913, 123084, 292784, 662512, 1434508, 2985639, 5997455, 11666253, 22040149, 40541549, 72770415, 127706638, 219494579, 370035418, 612722058, 997726253, 1599424522, 2526675993, 3936943281

Offset: 1

R. H. Hardin, Dec 07 2013

nonn

Row 3 of A233301, row 4 of A267245.

			Some solutions for n=5
..0..0..1..0..0..0....0..1..0..0..0..1....0..0..0..0..0..1....1..1..1..0..0..0
..1..0..0..1..0..0....0..0..0..1..1..1....1..0..1..0..0..1....0..0..0..1..1..1
..1..0..0..0..1..1....0..0..0..1..1..1....0..1..1..0..0..1....1..0..1..1..1..1
..0..1..1..1..1..1....0..0..1..1..1..1....0..0..0..1..1..1....0..1..1..1..1..1

R. H. Hardin, Table of n, a(n) for n = 1..66

A267240 Number of n X 3 binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

Original entry on oeis.org

4, 13, 42, 141, 486, 1685, 5804, 19769, 66544, 221581, 730918, 2391717, 7772610, 25110933, 80713016, 258280817, 823269116, 2615088973, 8281113730, 26150883901, 82375282494, 258893742933, 811984918692, 2541865829801, 7943330715176

Offset: 1

R. H. Hardin, Jan 12 2016

nonn

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted proof of empirical recurrence
Index entries for linear recurrences with constant coefficients, signature (10, -39, 76, -79, 42, -9).

Column 3 of A267245.

Empirical: a(n) = 10*a(n-1) - 39*a(n-2) + 76*a(n-3) - 79*a(n-4) + 42*a(n-5) - 9*a(n-6).
Conjectures from Colin Barker, Jan 11 2019: (Start)
G.f.: x*(4 - 27*x + 68*x^2 - 76*x^3 + 42*x^4 - 9*x^5) / ((1 - x)^4*(1 - 3*x)^2).
a(n) = (24 + (31+3^(2+n))*n + 12*n^2 + 2*n^3) / 24.
(End)
Empirical recurrence verified (see link). - Robert Israel, Sep 08 2019

A267241 Number of nX4 binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

Original entry on oeis.org

5, 22, 105, 567, 3351, 20676, 129129, 804817, 4982759, 30629206, 187121865, 1137631979, 6891047527, 41628865000, 250987078681, 1511105743781, 9088662549303, 54625229882746, 328144877989145, 1970524978549951

Offset: 1

R. H. Hardin, Jan 12 2016

nonn

Column 4 of A267245.

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted proof of empirical recurrence
Index entries for linear recurrences with constant coefficients, signature (24, -246, 1420, -5121, 12084, -18944, 19536, -12720, 4736, -768).

Cf. A267245.

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)]):
T:= Matrix(54,54,proc(i,j) local k;
  if add(states[j,k]-states[i,k],k=1..4) > 0 then return 0 fi;
  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;
  if states[i,1]>=states[i,2] and states[j,5]<> states[i,5] then return 0 fi;
  if states[i,2]>=states[i,3] and states[j,6]<> states[i,6] then return 0 fi;
  if states[i,3]>=states[i,4] and states[j,7]<> states[i,7] then return 0 fi;
1
end proc):
U:= Vector(54,1):
E[0]:= Vector(54): E[0][1]:= 1:
for k from 1 to 25 do E[k]:= T . E[k-1] od:
seq(U^%T . E[j], j=1..25); # Robert Israel, Sep 08 2019

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 *)

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).

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

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

Original entry on oeis.org

6, 34, 232, 1986, 20040, 220235, 2499080, 28501471, 323067002, 3626695952, 40306404192, 443852375808, 4848323701804, 52590398731297, 567018802063680, 6081537709403509, 64929807220896558, 690446673537426382

Offset: 1

R. H. Hardin, Jan 12 2016

nonn

Column 5 of A267245.

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

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 (52, -1196, 16140, -142918, 879116, -3875668, 12442580, -29232481, 50015232, -61355336, 52355680, -29405200, 9744000, -1440000).

Cf. A267245.

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 5 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[5]:
T:= Matrix(162,162,proc(i,j) local k;
  if add(states[j,k]-states[i,k],k=1..5) > 0 then return 0 fi;
  for k from 6 to 9 do if states[j,k]>states[i,k] then return 0 fi od;
  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;
1
end proc):
E:= Vector(162): E[1]:= 1:
U[0]:= Vector[row](162,1):
for k from 1 to 25 do U[k]:= U[k-1].T od:
seq(U[j] . E, j=1..25); # Robert Israel, Sep 08 2019

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).

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

A267243 Number of n X 6 binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

Original entry on oeis.org

7, 50, 475, 6292, 107015, 2093467, 43555569, 924051709, 19614050515, 413556580944, 8645774602327, 179276181587698, 3691120876565687, 75550095426967737, 1538986699132717645, 31229753343696948035

Offset: 1

R. H. Hardin, Jan 12 2016

nonn

Column 6 of A267245.

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

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted proof of empirical recurrence
Index entries for linear recurrences with constant coefficients, signature (114, -5915, 186008, -3982785, 61835542, -723657627, 6549515604, -46652032035, 264676225246, -1205477853945, 4427867737616, -13139368875011, 31468929403866, -60602488003009, 93197329064964, -113220771193368, 106920682204032, -76630180181904, 40173465734208, -14497964755200, 3213273369600, -329204736000).

Cf. A267245.

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 6 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[6]:
T:= Matrix(486,486,proc(i,j) local k;
  if add(states[j,k]-states[i,k],k=1..6) > 0 then return 0 fi;
  for k from 7 to 11 do if states[j,k]>states[i,k] then return 0 fi od;
  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;
1
end proc):
E:= Vector(486): E[1]:= 1:
U[0]:= Vector[row](486,1):
for k from 1 to 25 do U[k]:= U[k-1].T od:
seq(U[j] . E, j=1..25); # Robert Israel, Sep 08 2019

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).

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

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

Original entry on oeis.org

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

Offset: 1

R. H. Hardin, Jan 12 2016

nonn

Column 7 of A267245.

			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

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).

Cf. A267245.

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

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

A267239 Number of n X n binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

Original entry on oeis.org

2, 7, 42, 567, 20040, 2093467, 683027146, 723576702570, 2549735632656528, 30465608408449993885, 1250919952257037350168062, 178764247753852768666003981113, 89752660973687899647122735126286078

Offset: 1

R. H. Hardin, Jan 12 2016

nonn

Diagonal of A267245.

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

Cf. A267245.

A267248 Number of 5Xn binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

Original entry on oeis.org

6, 63, 486, 3351, 20040, 107015, 516084, 2278687, 9303052, 35428544, 126753732, 428651446, 1377310514, 4223642819, 12409437728, 35050866249, 95459908872, 251341796912, 641282264972, 1588863292072

Offset: 1

R. H. Hardin, Jan 12 2016

nonn

Row 5 of A267245.

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

Cf. A267245.

A267249 Number of 6Xn binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

Original entry on oeis.org

7, 127, 1685, 20676, 220235, 2093467, 17892539, 139164323, 993947185, 6572368601, 40512935649, 234192144667, 1276154088792, 6584891361478, 32302298031855, 151176059240006

Offset: 1

R. H. Hardin, Jan 12 2016

nonn

Row 6 of A267245.

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

Cf. A267245.

cp's OEIS Frontend

A233302 Number of (2+1) X (n+1) 0..1 arrays x(i,j) with row sums sum{x(i,j), j=1..n+1} nondecreasing, and column sums sum{i^2*x(i,j), i=1..2+1} nondecreasing.

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A233303 Number of (3+1)X(n+1) 0..1 arrays x(i,j) with row sums sum{x(i,j), j=1..n+1} nondecreasing, and column sums sum{i^2*x(i,j), i=1..3+1} nondecreasing.

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A267240 Number of n X 3 binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

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A267241 Number of nX4 binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

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

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A267243 Number of n X 6 binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

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

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Maple

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A267239 Number of n X n binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

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A267248 Number of 5Xn binary arrays with row sums nondecreasing and columns lexicographically nondecreasing.

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