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A283032
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Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to vertical and horizontal reflections.
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8
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0, 1, 8407040, 211829725395, 281475530358784, 74505821533203125, 7107572245840091136, 335267157313994232775, 9444732983468915425280, 179474497026544179696969, 2500000000502500000000000, 27086764860568185273201371, 238490541617902791488962560
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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 D2 acting on the 25 entries is (2*s(2)^10*s(1)^5 + s(2)^12*s(1) + s(1)^25)/4.
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LINKS
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FORMULA
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a(n) = n^13*(n^2 + 1)*(n^10 - n^8 + n^6 - n^4 + n^2 + 1)/4.
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)
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EXAMPLE
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For n=2 we get a(2)=8407040 inequivalent 5 X 5 binary matrices up to vertical and horizontal reflections.
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MAPLE
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[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
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MATHEMATICA
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Table[n^13*(n^2 + 1)*(n^10- n^8 + n^6 - n^4 + n^2 + 1)/4, {n, 0, 16}]
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PROG
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(PARI) a(n) = n^13*(n^2 + 1)*(n^10 - n^8 + n^6 - n^4 + n^2 + 1)/4; \\ Indranil Ghosh, Feb 27 2017
(Python) 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
(Magma) [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
(Sage) [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
(GAP) 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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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