|
%I #21 Sep 08 2022 08:46:18
%S 0,1,8407040,211829725395,281475530358784,74505821533203125,
%T 7107572245840091136,335267157313994232775,9444732983468915425280,
%U 179474497026544179696969,2500000000502500000000000,27086764860568185273201371,238490541617902791488962560
%N Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to vertical and horizontal reflections.
%C 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.
%H G. C. Greubel, <a href="/A283032/b283032.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = n^13*(n^2 + 1)*(n^10 - n^8 + n^6 - n^4 + n^2 + 1)/4.
%F From _Chai Wah Wu_, Dec 07 2018: (Start)
%F 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.
%F 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)
%e For n=2 we get a(2)=8407040 inequivalent 5 X 5 binary matrices up to vertical and horizontal reflections.
%p [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
%t Table[n^13*(n^2 + 1)*(n^10- n^8 + n^6 - n^4 + n^2 + 1)/4, {n, 0, 16}]
%o (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
%o (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
%o (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
%o (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
%o (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
%Y Cf. A282612, A282613, A283026, A283027, A283029, A283030, A283031, A283033.
%Y Cf. A283028 (4 X 4 version), A282614 (3 X 3 version), A039623 (2 X 2 version).
%K nonn,easy
%O 0,3
%A _David Nacin_, Feb 27 2017
| 