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%I #13 Dec 07 2018 14:49:44
%S 0,1,16793600,423651479175,562950490292224,149011627197265625,
%T 14215144250057342976,670534312205763205375,18889465949070766899200,
%U 358948993948871860432449,5000000000500000000000000,54173529719030485105622951,476981083228048575587942400
%N Number of inequivalent 5 X 5 matrices with entries in {1,2,3,..,n} when a matrix and its transpose are considered equivalent.
%C Cycle index of symmetric group S2 acting on the set of 25 entries is (s(2)^10*s(1)^5 + s(1)^25)/2.
%F a(n) = n^15*(n^2+1)*(n^8-n^6+n^4-n^2+1)/2.
%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 + 16793574*x^23 + 423214845900*x^22 + 551941009751074*x^21 + 134512557517054626*x^20 + 10522699609491808746*x^19 + 347912001753554722204*x^18 + 5696453728178627889150*x^17 + 50977946159336791604079*x^16 + 265857130683340877431996*x^15 + 842694350441988138095256*x^14 + 1667306282568523129263444*x^13 + 2089823554970188253479900*x^12 + 1667306282568523129263444*x^11 + 842694350441988138095256*x^10 + 265857130683340877431996*x^9 + 50977946159336791604079*x^8 + 5696453728178627889150*x^7 + 347912001753554722204*x^6 + 10522699609491808746*x^5 + 134512557517054626*x^4 + 551941009751074*x^3 + 423214845900*x^2 + 16793574*x + 1)/(x - 1)^26. (End)
%e For n=2 we get a(2)=16793600 inequivalent 5x5 binary matrices up to the action of transposition.
%t Table[n^15 (n^2 + 1) (n^8 - n^6 + n^4 - n^2 + 1)/2, {n, 0, 12}]
%o (PARI) a(n) = n^15*(n^2+1)*(n^8-n^6+n^4-n^2+1)/2; \\ _Indranil Ghosh_, Feb 27 2017
%o (Python) def A283029(n): return n**15*(n**2+1)*(n**8-n**6+n**4-n**2+1)/2 # _Indranil Ghosh_, Feb 27 2017
%Y Cf. A282612,A282613,A282614. A283026, A283027, A283028, A283030, A283031, A283032, A283033. A170798 (4x4 version). A168555 (3x3 version). A019582 (2x2 version)
%K nonn,easy
%O 0,3
%A _David Nacin_, Feb 27 2017