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%I #22 Sep 08 2022 08:46:18
%S 0,1,376992,7355513529,9474438804480,2491483056641250,
%T 237223883948569056,11182222570880983622,314920519245916176384,
%U 5983496429606726016735,83341666958337500020000,902948225666983587054711,7950004204832195461143552,58805000552467321853765064
%N Number of inequivalent 5 X 5 matrices with entries in {1,2,3,...,n} up to row permutations.
%C Cycle index of symmetric group S4 on the set of 25 entries is (10*s(2)^5*s(1)^15 + 20*s(3)^5*s(1)^10 + 15*s(2)^10*s(1)^5 + 30*s(4)^5*s(1)^5 + 20*s(2)^5*s(3)^5 + 24*s(5)^5+s(1)^25)/120.
%H G. C. Greubel, <a href="/A283030/b283030.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = (n^25 + 10*n^20 + 35*n^15 + 50*n^10 + 24*n^5)/120.
%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*(201376*x^23 + 6769097812*x^22 + 9115118766616*x^21 + 2236218775591321*x^20 + 175251248958400030*x^19 + 5797456665826176046*x^18 + 94937993285056078902*x^17 + 849635569433212953261*x^16 + 4430970723887327210136*x^15 + 14044903652456409705760*x^14 + 27788396155245137222056*x^13 + 34830392581327241688322*x^12 + 27788479931754180338596*x^11 + 14044908029988217540516*x^10 + 4430933630938187561140*x^9 + 849629302807069561746*x^8 + 94943797840269544152*x^7 + 5799609980863901436*x^6 + 175505398388141776*x^5 + 2247537209457445*x^4 + 9283317972526*x^3 + 7345712062*x^2 + 376966*x + 1)/(x - 1)^26. (End)
%e For n=2 we get a(2)=376992 inequivalent 5 X 5 binary matrices up to row permutations.
%p [(10*n^20+35*n^15+50*n^10+24*n^5+n^25)/120$n=0..16]; # _Muniru A Asiru_, Dec 07 2018
%t Table[(10n^20+ 35n^15 + 50n^10 + 24n^5 + n^25)/120, {n, 0, 16}]
%o (PARI) a(n) = (10*n^20 + 35*n^15 + 50*n^10 + 24*n^5 + n^25)/120; \\ _Indranil Ghosh_, Feb 27 2017
%o (Python) def A283030(n): return (10*n**20 + 35*n**15 + 50*n**10 + 24*n**5 + n**25)/120 # _Indranil Ghosh_, Feb 27 2017
%o (Magma) [n^5*(n^20 +10*n^15 +35*n^10 +50*n^5 +24)/120: n in [0..20]]; // _G. C. Greubel_, Dec 07 2018
%o (Sage) [n^5*(n^20 +10*n^15 +35*n^10 +50*n^5 +24)/120 for n in range(20)] # _G. C. Greubel_, Dec 07 2018
%o (GAP) List([0..20], n -> n^5*(n^20 +10*n^15 +35*n^10 +50*n^5 +24)/120); # _G. C. Greubel_, Dec 07 2018
%Y Cf. A282613, A282614, A283027, A283028, A283029, A283031, A283032, A283033.
%Y Cf. A283026 (4 X 4 version), A282612 (3 X 3 version), A037270 (2 X 2 version).
%K nonn,easy
%O 0,3
%A _David Nacin_, Feb 27 2017
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