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

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

Offset: 0

Author

David Nacin, Feb 27 2017

Keywords

• nonn
• easy

Comments

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).

Examples

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

Links

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

Crossrefs

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).

Programs

GAP

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

Magma

[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

Maple

[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

Mathematica

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

PARI

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

Python

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

Python

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

Sage

[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

Formula

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