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A283033
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Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to rotations and reflections.
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9
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0, 1, 4211744, 105918450471, 140738033618944, 37252918396015625, 3553786240466361696, 167633579843887699759, 4722366500530551259136, 89737248564744874067889, 1250000000501250002500000, 13543382431328404683826391, 119245270812803151147085824
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = n^7*(n^18 + 4*n^8 + n^6 + 2)/8.
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)
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EXAMPLE
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For n=2 we get a(2)=4211744 inequivalent 5 X 5 binary matrices up to rotations and reflections.
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MAPLE
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MATHEMATICA
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Table[n^7 (n^18 + 4 n^8 + n^6 + 2)/8, {n, 0, 16}]
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PROG
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(PARI) a(n) = 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
(Sage) [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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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