A282614 -id:A282614 - cp's OEIS frontend

A282613 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to rotations.

0, 1, 140, 4995, 65824, 489125, 2521476, 10092775, 33562880, 96870249, 250025500, 589527851, 1290008160, 2651218765, 5165397524, 9611031375, 17180133376, 29647326545, 49590297900, 80672546899, 128000804000, 198571037301, 301818598180, 450289780535

Offset: 0

David Nacin, Feb 19 2017

Cycle index of symmetry group (cyclic rotation group of order 4 acting on the 9 cells of the square) is (2s(4)^2*s(1) + s(2)^4*s(1) + s(1)^9)/4.

			The number of 3 X 3 binary matrices up to rotations is 140.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Row n=3 of A343095.
Cf. A282612, A282614, A217331, A168555.
Cf. A006528 (2 x 2 version), A283027 (4 X 4 version).

Mathematica
```
Table[(2n^3+n^5+n^9)/4, {n, 0, 24}]
```

concat(0, Vec(x*(1 + 130*x + 3640*x^2 + 22054*x^3 + 39070*x^4 + 22054*x^5 + 3640*x^6 + 130*x^7 + x^8) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Feb 23 2017

a(n) = n^3*(n^2+1)*(n^4-n^2+2)/4.

G.f.: x*(1 + 130*x + 3640*x^2 + 22054*x^3 + 39070*x^4 + 22054*x^5 + 3640*x^6 + 130*x^7 + x^8) / (1 - x)^10. - Colin Barker, Feb 23 2017

A282612 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to row permutations.

Original entry on oeis.org

0, 1, 120, 3654, 45760, 333375, 1703016, 6784540, 22500864, 64836045, 167167000, 393877506, 861456960, 1769830699, 3447273480, 6412923000, 11461636096, 19776716505, 33076889784, 53804808190, 85365336000, 132422893911, 201268229800, 300266132244, 440396812800

Offset: 0

David Nacin, Feb 19 2017

Cycle index of symmetry group is (3*s(2)^3*s(1)^3 + 2*s(3)^3 + s(1)^9)/6.

			The number of 3 X 3 binary matrices up to row permutations is 120.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Cf. A282613, A282614, A217331, A168555. A037270 (2x2 version.)

Mathematica
```
Table[(3n^6+2n^3+n^9)/6, {n, 0, 24}]
```

concat(0, Vec(x*(1 + 110*x + 2499*x^2 + 14500*x^3 + 26015*x^4 + 14934*x^5 + 2365*x^6 + 56*x^7) / (1 - x)^10 + O(x^30))) \\ Colin Barker, Feb 23 2017

a(n) = n^3*(n^3+2)*(n+1)*(n^2-n+1)/6.

G.f.: x*(1 + 110*x + 2499*x^2 + 14500*x^3 + 26015*x^4 + 14934*x^5 + 2365*x^6 + 56*x^7) / (1 - x)^10. - Colin Barker, Feb 23 2017

A283027 Number of inequivalent 4 X 4 matrices with entries in {1,2,3,...,n} up to rotations.

Original entry on oeis.org

0, 1, 16456, 10763361, 1073758336, 38147070625, 705277897416, 8308234084801, 70368748374016, 463255057977921, 2500000025005000, 11487432519490081, 46221064831264896, 166354152499741921, 544488334821309256, 1642102089568970625, 4611686019501162496

Offset: 0

David Nacin, Feb 27 2017

Cycle index of cyclic group C4 acting on the 16 entries is (2s(4)^4 + s(2)^8 + s(1)^16)/24.

			For n=2 we get a(2)=16456 inequivalent 4 X 4 binary matrices up to rotations.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (17,-136,680,-2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380,-680,136,-17,1).

Row n=4 of A343095.
Cf. A282612, A282614, A283026, A283028, A283029, A283030, A283032, A283033.
Cf. A283031 (5 X 5 version), A282613 (3 X 3 version), A006528 (2 X 2 version).

List([0..20], n -> n^4*(n^4+1)*(n^8-n^4+2)/4); # G. C. Greubel, Dec 07 2018

[n^4*(n^4+1)*(n^8-n^4+2)/4: n in [0..20]]; // G. C. Greubel, Dec 07 2018

[n^4*(n^4+1)*(n^8-n^4+2)/4$n=0..18]; # Muniru A Asiru, Dec 07 2018

Table[n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4,{n,0,30}]

a(n) = n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4; \\ Indranil Ghosh, Feb 27 2017

def A283027(n) : return n**4*(n**4 + 1)*(n**8 - n**4 + 2)/4 # Indranil Ghosh, Feb 27 2017

def A283027(n): return n**4*(n**4*(n**8+1)+2)>>2 # Chai Wah Wu, Jan 24 2023

[n^4*(n^4+1)*(n^8-n^4+2)/4 for n in range(20)] # G. C. Greubel, Dec 07 2018

a(n) = n^4*(n^4 + 1)*(n^8 - n^4 + 2)/4.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 17*a(n-1) - 136*a(n-2) + 680*a(n-3) - 2380*a(n-4) + 6188*a(n-5) - 12376*a(n-6) + 19448*a(n-7) - 24310*a(n-8) + 24310*a(n-9) - 19448*a(n-10) + 12376*a(n-11) - 6188*a(n-12) + 2380*a(n-13) - 680*a(n-14) + 136*a(n-15) - 17*a(n-16) + a(n-17) for n > 16.
G.f.: -x*(x + 1)*(x^14 + 16438*x^13 + 10467307*x^12 + 882551228*x^11 + 20463257081*x^10 + 175065647018*x^9 + 626805100059*x^8 + 968894657736*x^7 + 626805100059*x^6 + 175065647018*x^5 + 20463257081*x^4 + 882551228*x^3 + 10467307*x^2 + 16438*x + 1)/(x - 1)^17. (End)
E.g.f.: (1/4)*x*(x^15 + 120*x^14 + 6020*x^13 + 165620*x^12 + 2757118*x^11 + 28936908*x^10 + 193754990*x^9 + 820784250*x^8 + 2141764054*x^7 + 3281882632*x^6 + 2734926824*x^5 + 1096191600*x^4 + 171800604*x^3 + 7142664*x^2 + 32908*x + 4)*exp(x). - Stefano Spezia, Dec 07 2018
a(n) = n^4*(n^12 + n^4 + 2)/4. - Chai Wah Wu, Jan 24 2023

A283031 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to rotations.

Original entry on oeis.org

0, 1, 8390720, 211822552035, 281474993496064, 74505806274453125, 7107572010747738816, 335267154940213889575, 9444732965876730429440, 179474496923598616041129, 2500000000002500005000000, 27086764858479561198237131, 238490541610199280719585280

Offset: 0

David Nacin, Feb 27 2017

Cycle index of cyclic group C4 acting on the set of 25 entries is (2*s(4)^6*s(1) + s(2)^12*s(1) + s(1)^25).

			For n=2 we get a(2)=8390720 inequivalent 5 X 5 binary matrices up to rotations.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Row n=5 of A343095.
Cf. A282612, A282614, A283028, A283029, A283030, A283032, A283033.
Cf. A283027 (4 X 4 version), A282613 (3 X 3 version), A006528 (2 X 2 version).

List([0..30], n -> n^7*(n^2+1)*(n^4-n^2+1)*(n^12-n^6+2)/4); # G. C. Greubel, Dec 07 2018

[n^7*(n^2+1)*(n^4-n^2+1)*(n^12-n^6+2)/4: n in [0..20]]; // G. C. Greubel, Dec 07 2018

[n^7*(n^2+1)*(n^4-n^2+1)*(n^12-n^6+2)/4$n=0..16]; # Muniru A Asiru, Dec 07 2018

Table[n^7(n^2 + 1)(n^4 - n^2 + 1)(n^12 - n^6 + 2)/4, {n, 0, 16}]

a(n) = n^7*(n^2 + 1)*(n^4 - n^2 + 1)*(n^12 - n^6 + 2)/4; \\ Indranil Ghosh, Feb 27 2017

def A283031(n): return n**7*(n**2 + 1)*(n**4 - n**2 + 1)*(n**12 - n**6 + 2)/4 # Indranil Ghosh, Feb 27 2017

def A283031(n): return n**7*(n**6*(n**12+1)+2)>>2 # Chai Wah Wu, Jan 24 2023

[n^7*(n^2+1)*(n^4-n^2+1)*(n^12-n^6+2)/4 for n in range(20)] # G. C. Greubel, Dec 07 2018

a(n) = n^7*(n^2 + 1)*(n^4 - n^2 + 1)*(n^12 - n^6 + 2)/4.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 26*a(n-1) - 325*a(n-2) + 2600*a(n-3) - 14950*a(n-4) + 65780*a(n-5) - 230230*a(n-6) + 657800*a(n-7) - 1562275*a(n-8) + 3124550*a(n-9) - 5311735*a(n-10) + 7726160*a(n-11) - 9657700*a(n-12) + 10400600*a(n-13) - 9657700*a(n-14) + 7726160*a(n-15) - 5311735*a(n-16) + 3124550*a(n-17) - 1562275*a(n-18) + 657800*a(n-19) - 230230*a(n-20) + 65780*a(n-21) - 14950*a(n-22) + 2600*a(n-23) - 325*a(n-24) + 26*a(n-25) - a(n-26) for n > 25.
G.f.: x*(x^24 + 8390694*x^23 + 211604393640*x^22 + 275970334124554*x^21 + 67256276957109786*x^20 + 5261349807304085586*x^19 + 173956000912091771464*x^18 + 2848226864007694392990*x^17 + 25488973079546662159119*x^16 + 132928565342248912495516*x^15 + 421347175220529448574736*x^14 + 833653141283634765151044*x^13 + 1044911777486454930701740*x^12 + 833653141283634765151044*x^11 + 421347175220529448574736*x^10 + 132928565342248912495516*x^9 + 25488973079546662159119*x^8 + 2848226864007694392990*x^7 + 173956000912091771464*x^6 + 5261349807304085586*x^5 + 67256276957109786*x^4 + 275970334124554*x^3 + 211604393640*x^2 + 8390694*x + 1)/(x - 1)^26. (End)
a(n) = n^7*(n^18 + n^6 + 2)/4. - Chai Wah Wu, Jan 24 2023

A283033 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to rotations and reflections.

Original entry on oeis.org

0, 1, 4211744, 105918450471, 140738033618944, 37252918396015625, 3553786240466361696, 167633579843887699759, 4722366500530551259136, 89737248564744874067889, 1250000000501250002500000, 13543382431328404683826391, 119245270812803151147085824

Offset: 0

David Nacin, Feb 27 2017

nonn

Cycle index of dihedral group D4 acting on the 25 entries is (2*s(4)^6*s(1) + s(2)^{12}*s(1) + 4*s(2)^10*s(1)^5 + s(1)^25)/8.

			For n=2 we get a(2)=4211744 inequivalent 5 X 5 binary matrices up to rotations and reflections.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Row n=5 of A343097.
Cf. A282612, A282613, A282614, A283026, A283027, A283028, A283029, A283030, A283031, A283032.
Cf. A217338 (4 X 4 version), A217331 (3 X 3 version), A002817 (2 X 2 version).

List([0..20], n -> n^7*(n^18+4*n^8+n^6+2)/8); # G. C. Greubel, Dec 07 2018

[n^7*(n^18+4*n^8+n^6+2)/8: n in [0..20]]; // G. C. Greubel, Dec 07 2018

[n^7*(n^18+4*n^8+n^6+2)/8$n=0..16]; # Muniru A Asiru, Dec 07 2018

Table[n^7 (n^18 + 4 n^8 + n^6 + 2)/8, {n, 0, 16}]

a(n) = n^7*(n^18 + 4*n^8 + n^6 + 2)/8; \\ Indranil Ghosh, Feb 27 2017

def A283033(n): return n**7*(n**18 + 4*n**8 + n**6 + 2)/8 # Indranil Ghosh, Feb 27 2017

[n^7*(n^18+4*n^8+n^6+2)/8 for n in range(20)] # G. C. Greubel, Dec 07 2018

a(n) = n^7*(n^18 + 4*n^8 + n^6 + 2)/8.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 26*a(n-1) - 325*a(n-2) + 2600*a(n-3) - 14950*a(n-4) + 65780*a(n-5) - 230230*a(n-6) + 657800*a(n-7) - 1562275*a(n-8) + 3124550*a(n-9) - 5311735*a(n-10) + 7726160*a(n-11) - 9657700*a(n-12) + 10400600*a(n-13) - 9657700*a(n-14) + 7726160*a(n-15) - 5311735*a(n-16) + 3124550*a(n-17) - 1562275*a(n-18) + 657800*a(n-19) - 230230*a(n-20) + 65780*a(n-21) - 14950*a(n-22) + 2600*a(n-23) - 325*a(n-24) + 26*a(n-25) - a(n-26) for n > 25.
G.f.: x*(x^24 + 4211718*x^23 + 105808945452*x^22 + 137985522720898*x^21 + 33628142067806706*x^20 + 2630674898090394666*x^19 + 86978000386844370748*x^18 + 1424113432167998385342*x^17 + 12744486540004851097263*x^16 + 66464282669989885009756*x^15 + 210673587611186802329496*x^14 + 416826570643036689533748*x^13 + 522455888740564118388412*x^12 + 416826570643036689533748*x^11 + 210673587611186802329496*x^10 + 66464282669989885009756*x^9 + 12744486540004851097263*x^8 + 1424113432167998385342*x^7 + 86978000386844370748*x^6 + 2630674898090394666*x^5 + 33628142067806706*x^4 + 137985522720898*x^3 + 105808945452*x^2 + 4211718*x + 1)/(x - 1)^26. (End)

A283028 Number of inequivalent 4 X 4 matrices with entries in {1,2,3,...,n} up to vertical and horizontal reflections.

Original entry on oeis.org

0, 1, 16576, 10766601, 1073790976, 38147265625, 705278736576, 8308236966001, 70368756760576, 463255079498001, 2500000075000000, 11487432626662201, 46221065046245376, 166354152907593001, 544488335559184576, 1642102090850390625, 4611686021648613376

Offset: 0

David Nacin, Feb 27 2017

Cycle index of dihedral group D2 acting on the 16 entries is (3s(2)^8 + s(1)^16)/4.

			For n=2 we get a(2)=16576 inequivalent 4 X 4 binary matrices up to vertical and horizontal reflections.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. A282612, A282613, A283026, A283027, A283029, A283030, A283031, A283033.

Cf. A283032 (5 X 5 version), A282614 (3 X 3 version), A039623 (2 X 2 version).

List([0..20], n -> n^8*(n^8+3)/4); # G. C. Greubel, Dec 07 2018

[n^8*(n^8+3)/4: n in [0..20]]; // G. C. Greubel, Dec 07 2018

[n^8*(n^8+3)/4$n=0..18]; # Muniru A Asiru, Dec 07 2018

Mathematica
```
Table[n^8(n^8 + 3)/4,{n,0,30}]
```

a(n) = n^8 * (n^8 + 3)/4; \\ Altug Alkan, Feb 27 2017

def A283028(n): return n**8*(n**8 + 3)/4 # Indranil Ghosh, Feb 27 2017

[n^8*(n^8+3)/4 for n in range(20)] # G. C. Greubel, Dec 07 2018

a(n) = n^8 * (n^8 + 3)/4.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 17*a(n-1) - 136*a(n-2) + 680*a(n-3) - 2380*a(n-4) + 6188*a(n-5) - 12376*a(n-6) + 19448*a(n-7) - 24310*a(n-8) + 24310*a(n-9) - 19448*a(n-10) + 12376*a(n-11) - 6188*a(n-12) + 2380*a(n-13) - 680*a(n-14) + 136*a(n-15) - 17*a(n-16) + a(n-17) for n > 16.
G.f.: -x*(x + 1)*(x^14 + 16558*x^13 + 10468387*x^12 + 882544028*x^11 + 20463263441*x^10 + 175065686258*x^9 + 626804969739*x^8 + 968894839176*x^7 + 626804969739*x^6 + 175065686258*x^5 + 20463263441*x^4 + 882544028*x^3 + 10468387*x^2 + 16558*x + 1)/(x - 1)^17. (End)
E.g.f.: (1/4)*x*exp(x)*(x^15 + 120*x^14 + 6020*x^13 + 165620*x^12 + 2757118*x^11 + 28936908*x^10 + 193754990*x^9 + 820784250*x^8 + 2141764056*x^7 + 3281882688*x^6 + 2734927356*x^5 + 1096193700*x^4 + 171804004*x^3 + 7144584*x^2 + 33148*x + 4). - Stefano Spezia, Dec 07 2018

A283029 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,..,n} when a matrix and its transpose are considered equivalent.

Original entry on oeis.org

0, 1, 16793600, 423651479175, 562950490292224, 149011627197265625, 14215144250057342976, 670534312205763205375, 18889465949070766899200, 358948993948871860432449, 5000000000500000000000000, 54173529719030485105622951, 476981083228048575587942400

Offset: 0

David Nacin, Feb 27 2017

Cycle index of symmetric group S2 acting on the set of 25 entries is (s(2)^10*s(1)^5 + s(1)^25)/2.

			For n=2 we get a(2)=16793600 inequivalent 5x5 binary matrices up to the action of transposition.

Cf. A282612,A282613,A282614. A283026, A283027, A283028, A283030, A283031, A283032, A283033. A170798 (4x4 version). A168555 (3x3 version). A019582 (2x2 version)

Table[n^15 (n^2 + 1) (n^8 - n^6 + n^4 - n^2 + 1)/2, {n, 0, 12}]

a(n) = n^15*(n^2+1)*(n^8-n^6+n^4-n^2+1)/2; \\ Indranil Ghosh, Feb 27 2017

def A283029(n): return n**15*(n**2+1)*(n**8-n**6+n**4-n**2+1)/2 # Indranil Ghosh, Feb 27 2017

a(n) = n^15*(n^2+1)*(n^8-n^6+n^4-n^2+1)/2.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 26*a(n-1) - 325*a(n-2) + 2600*a(n-3) - 14950*a(n-4) + 65780*a(n-5) - 230230*a(n-6) + 657800*a(n-7) - 1562275*a(n-8) + 3124550*a(n-9) - 5311735*a(n-10) + 7726160*a(n-11) - 9657700*a(n-12) + 10400600*a(n-13) - 9657700*a(n-14) + 7726160*a(n-15) - 5311735*a(n-16) + 3124550*a(n-17) - 1562275*a(n-18) + 657800*a(n-19) - 230230*a(n-20) + 65780*a(n-21) - 14950*a(n-22) + 2600*a(n-23) - 325*a(n-24) + 26*a(n-25) - a(n-26) for n > 25.
G.f.: x*(x^24 + 16793574*x^23 + 423214845900*x^22 + 551941009751074*x^21 + 134512557517054626*x^20 + 10522699609491808746*x^19 + 347912001753554722204*x^18 + 5696453728178627889150*x^17 + 50977946159336791604079*x^16 + 265857130683340877431996*x^15 + 842694350441988138095256*x^14 + 1667306282568523129263444*x^13 + 2089823554970188253479900*x^12 + 1667306282568523129263444*x^11 + 842694350441988138095256*x^10 + 265857130683340877431996*x^9 + 50977946159336791604079*x^8 + 5696453728178627889150*x^7 + 347912001753554722204*x^6 + 10522699609491808746*x^5 + 134512557517054626*x^4 + 551941009751074*x^3 + 423214845900*x^2 + 16793574*x + 1)/(x - 1)^26. (End)

A283030 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to row permutations.

Original entry on oeis.org

0, 1, 376992, 7355513529, 9474438804480, 2491483056641250, 237223883948569056, 11182222570880983622, 314920519245916176384, 5983496429606726016735, 83341666958337500020000, 902948225666983587054711, 7950004204832195461143552, 58805000552467321853765064

Offset: 0

David Nacin, Feb 27 2017

Cycle index of symmetric group S4 on the set of 25 entries is (10*s(2)^5*s(1)^15 + 20*s(3)^5*s(1)^10 + 15*s(2)^10*s(1)^5 + 30*s(4)^5*s(1)^5 + 20*s(2)^5*s(3)^5 + 24*s(5)^5+s(1)^25)/120.

			For n=2 we get a(2)=376992 inequivalent 5 X 5 binary matrices up to row permutations.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A282613, A282614, A283027, A283028, A283029, A283031, A283032, A283033.

Cf. A283026 (4 X 4 version), A282612 (3 X 3 version), A037270 (2 X 2 version).

List([0..20], n -> n^5*(n^20 +10*n^15 +35*n^10 +50*n^5 +24)/120); # G. C. Greubel, Dec 07 2018

[n^5*(n^20 +10*n^15 +35*n^10 +50*n^5 +24)/120: n in [0..20]]; // G. C. Greubel, Dec 07 2018

[(10*n^20+35*n^15+50*n^10+24*n^5+n^25)/120$n=0..16]; # Muniru A Asiru, Dec 07 2018

Table[(10n^20+ 35n^15 + 50n^10 + 24n^5 + n^25)/120, {n, 0, 16}]

a(n) = (10*n^20 + 35*n^15 + 50*n^10 + 24*n^5 + n^25)/120; \\ Indranil Ghosh, Feb 27 2017

def A283030(n): return (10*n**20 + 35*n**15 + 50*n**10 + 24*n**5 + n**25)/120 # Indranil Ghosh, Feb 27 2017

[n^5*(n^20 +10*n^15 +35*n^10 +50*n^5 +24)/120 for n in range(20)] # G. C. Greubel, Dec 07 2018

a(n) = (n^25 + 10*n^20 + 35*n^15 + 50*n^10 + 24*n^5)/120.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 26*a(n-1) - 325*a(n-2) + 2600*a(n-3) - 14950*a(n-4) + 65780*a(n-5) - 230230*a(n-6) + 657800*a(n-7) - 1562275*a(n-8) + 3124550*a(n-9) - 5311735*a(n-10) + 7726160*a(n-11) - 9657700*a(n-12) + 10400600*a(n-13) - 9657700*a(n-14) + 7726160*a(n-15) - 5311735*a(n-16) + 3124550*a(n-17) - 1562275*a(n-18) + 657800*a(n-19) - 230230*a(n-20) + 65780*a(n-21) - 14950*a(n-22) + 2600*a(n-23) - 325*a(n-24) + 26*a(n-25) - a(n-26) for n > 25.
G.f.: x*(201376*x^23 + 6769097812*x^22 + 9115118766616*x^21 + 2236218775591321*x^20 + 175251248958400030*x^19 + 5797456665826176046*x^18 + 94937993285056078902*x^17 + 849635569433212953261*x^16 + 4430970723887327210136*x^15 + 14044903652456409705760*x^14 + 27788396155245137222056*x^13 + 34830392581327241688322*x^12 + 27788479931754180338596*x^11 + 14044908029988217540516*x^10 + 4430933630938187561140*x^9 + 849629302807069561746*x^8 + 94943797840269544152*x^7 + 5799609980863901436*x^6 + 175505398388141776*x^5 + 2247537209457445*x^4 + 9283317972526*x^3 + 7345712062*x^2 + 376966*x + 1)/(x - 1)^26. (End)

A283032 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to vertical and horizontal reflections.

Original entry on oeis.org

0, 1, 8407040, 211829725395, 281475530358784, 74505821533203125, 7107572245840091136, 335267157313994232775, 9444732983468915425280, 179474497026544179696969, 2500000000502500000000000, 27086764860568185273201371, 238490541617902791488962560

Offset: 0

David Nacin, Feb 27 2017

Cycle index of dihedral group D2 acting on the 25 entries is (2*s(2)^10*s(1)^5 + s(2)^12*s(1) + s(1)^25)/4.

			For n=2 we get a(2)=8407040 inequivalent 5 X 5 binary matrices up to vertical and horizontal reflections.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A282612, A282613, A283026, A283027, A283029, A283030, A283031, A283033.

Cf. A283028 (4 X 4 version), A282614 (3 X 3 version), A039623 (2 X 2 version).

List([0..30], n -> n^13*(n^2+1)*(n^10-n^8+n^6-n^4+n^2+1)/4); # G. C. Greubel, Dec 07 2018

[n^13*(n^2+1)*(n^10-n^8+n^6-n^4+n^2+1)/4: n in [0..20]]; // G. C. Greubel, Dec 07 2018

[n^13*(n^2+1)*(n^10-n^8+n^6-n^4+n^2+1)/4$n=0..16]; # Muniru A Asiru, Dec 07 2018

Table[n^13*(n^2 + 1)*(n^10- n^8 + n^6 - n^4 + n^2 + 1)/4, {n, 0, 16}]

a(n) = n^13*(n^2 + 1)*(n^10 - n^8 + n^6 - n^4 + n^2 + 1)/4; \\ Indranil Ghosh, Feb 27 2017

def A283032(n): return n**13*(n**2 + 1)*(n**10 - n**8 + n**6 - n**4 + n**2 + 1)/4 # Indranil Ghosh, Feb 27 2017

[n^13*(n^2+1)*(n^10-n^8+n^6-n^4+n^2+1)/4 for n in range(20)] # G. C. Greubel, Dec 07 2018

a(n) = n^13*(n^2 + 1)*(n^10 - n^8 + n^6 - n^4 + n^2 + 1)/4.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 26*a(n-1) - 325*a(n-2) + 2600*a(n-3) - 14950*a(n-4) + 65780*a(n-5) - 230230*a(n-6) + 657800*a(n-7) - 1562275*a(n-8) + 3124550*a(n-9) - 5311735*a(n-10) + 7726160*a(n-11) - 9657700*a(n-12) + 10400600*a(n-13) - 9657700*a(n-14) + 7726160*a(n-15) - 5311735*a(n-16) + 3124550*a(n-17) - 1562275*a(n-18) + 657800*a(n-19) - 230230*a(n-20) + 65780*a(n-21) - 14950*a(n-22) + 2600*a(n-23) - 325*a(n-24) + 26*a(n-25) - a(n-26) for n > 25.
G.f.: x*(x^24 + 8407014*x^23 + 211611142680*x^22 + 275970689783914*x^21 + 67256280546339066*x^20 + 5261349801742569906*x^19 + 173956000842889834744*x^18 + 2848226864171846366430*x^17 + 25488973079778181586319*x^16 + 132928565341114340101276*x^15 + 421347175221451531355376*x^14 + 833653141284854063434884*x^13 + 1044911777483791594156780*x^12 + 833653141284854063434884*x^11 + 421347175221451531355376*x^10 + 132928565341114340101276*x^9 + 25488973079778181586319*x^8 + 2848226864171846366430*x^7 + 173956000842889834744*x^6 + 5261349801742569906*x^5 + 67256280546339066*x^4 + 275970689783914*x^3 + 211611142680*x^2 + 8407014*x + 1)/(x - 1)^26. (End)

A283026 Number of inequivalent 4 X 4 matrices with entries in {1,2,3,..,n} up to row permutations.

Original entry on oeis.org

0, 1, 3876, 1929501, 183181376, 6419043125, 118091211876, 1388168405001, 11745311589376, 77279801651001, 416916712502500, 1915356782994501, 7705740009485376, 27731516944463501, 90762229896563876, 273716119247180625, 768684707117285376, 2027695320242670001

Offset: 0

David Nacin, Feb 27 2017

Cycle index of symmetry group S4 acting on the 16 entries is (6*s(2)^4s(1)^8 + 8*s(3)^4s(1)^4 + 3*s(2)^8 + 6*s(4)^4 + s(1)^{16})/24.

			For n=2 we get a(2)=3876 inequivalent 4x4 binary matrices up to row permutations.

Cf. A282613, A282614, A283027, A283028, A283029, A283031, A283032, A283033. A283030 (5x5 version). A282612 (3x3 version). A037270 (2x2 version).

Table[n^4*(n^4 + 1)*(n^4 + 2)*(n^4 + 3)/24,{n,0,30}]

a(n) = n^4*(n^4 + 1)*(n^4 + 2)*(n^4 + 3)/24; \\ Indranil Ghosh, Feb 27 2017

def A283026(n) : return n**4*(n**4 + 1)*(n**4 + 2)*(n**4 + 3)/24 # Indranil Ghosh, Feb 27 2017

a(n) = n^4*(n^4 + 1)*(n^4 + 2)*(n^4 + 3)/24.
From Chai Wah Wu, Dec 07 2018: (Start)
a(n) = 17*a(n-1) - 136*a(n-2) + 680*a(n-3) - 2380*a(n-4) + 6188*a(n-5) - 12376*a(n-6) + 19448*a(n-7) - 24310*a(n-8) + 24310*a(n-9) - 19448*a(n-10) + 12376*a(n-11) - 6188*a(n-12) + 2380*a(n-13) - 680*a(n-14) + 136*a(n-15) - 17*a(n-16) + a(n-17) for n > 16.
G.f.: -x*(x + 1)*(x^14 + 3858*x^13 + 1859887*x^12 + 149046428*x^11 + 3415692141*x^10 + 29161611758*x^9 + 104450960739*x^8 + 161533106376*x^7 + 104450960739*x^6 + 29161611758*x^5 + 3415692141*x^4 + 149046428*x^3 + 1859887*x^2 + 3858*x + 1)/(x - 1)^17. (End)

cp's OEIS Frontend

A282613 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to rotations.

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A282612 Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to row permutations.

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A283027 Number of inequivalent 4 X 4 matrices with entries in {1,2,3,...,n} up to rotations.

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A283031 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to rotations.

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A283033 Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to rotations and reflections.

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A283028 Number of inequivalent 4 X 4 matrices with entries in {1,2,3,...,n} up to vertical and horizontal reflections.

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