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

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

Offset

0,3

Comments

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.

Links

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

Formula

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)

Example

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

Maple

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

Mathematica

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

Prog

(PARI) a(n) = n^7*(n^18 + 4*n^8 + n^6 + 2)/8; \\ Indranil Ghosh, Feb 27 2017
(Python) def A283033(n): return n**7*(n**18 + 4*n**8 + n**6 + 2)/8 # Indranil Ghosh, Feb 27 2017
(Magma) [n^7*(n^18+4*n^8+n^6+2)/8: n in [0..20]]; // G. C. Greubel, Dec 07 2018
(SageMath) [n^7*(n^18+4*n^8+n^6+2)/8 for n in range(20)] # G. C. Greubel, Dec 07 2018
(GAP) List([0..20], n -> n^7*(n^18+4*n^8+n^6+2)/8); # G. C. Greubel, Dec 07 2018

Crossrefs

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).

Sequence in context: A258884 A250543 A251800 * A209395 A159714 A232351

Adjacent sequences: A283030 A283031 A283032 * A283034 A283035 A283036

Keyword

nonn

Author

David Nacin, Feb 27 2017

Status

approved