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%I M3949 #45 Feb 28 2023 21:09:28
%S 1,5,28,156,863,4571,22952,108182,477136,1969270,7625579,27804973,
%T 95858868,313747418,978734539,2920530663,8363945469,23057872913,
%U 61357278239,157985305473,394486861086,957156158394,2260761331227
%N Number of 5-covers of an unlabeled n-set.
%C Number of 5 X n binary matrices with at least one 1 in every column up to row and column permutations. - _Andrew Howroyd_, Feb 28 2023
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Stefano Spezia, <a href="/A005785/b005785.txt">Table of n, a(n) for n = 0..10000</a>
%H R. J. Clarke, <a href="http://dx.doi.org/10.1016/0012-365X(90)90146-9">Covering a set by subsets</a>, Discrete Math., 81 (1990), 147-152.
%H Vladeta Jovovic, <a href="/A005748/a005748.pdf">Binary matrices up to row and column permutations</a>
%H <a href="/index/Rec#order_99">Index entries for linear recurrences with constant coefficients</a>, signature (7, -17, 13, 7, -11, 16, -44, 43, -65, 82, 46, -36, -28, -175, 85, -168, 504, 91, 77, -394, -664, -52, -382, 1642, 600, 813, -1209, -1632, -1650, -1050, 2982, 2124, 3592, -1360, -2074, -5329, -3607, 1970, 3608, 7640, 1778, 426, -8168, -6638, -3524, 2095, 8401, 6077, 5907, -5907, -6077, -8401, -2095, 3524, 6638, 8168, -426, -1778, -7640, -3608, -1970, 3607, 5329, 2074, 1360, -3592, -2124, -2982, 1050, 1650, 1632, 1209, -813, -600, -1642, 382, 52, 664, 394, -77, -91, -504, 168, -85, 175, 28, 36, -46, -82, 65, -43, 44, -16, 11, -7, -13, 17, -7, 1).
%F G.f.: - (x^68 - 2*x^67 + 10*x^66 + 32*x^65 + 175*x^64 + 794*x^63 + 3441*x^62 + 13186*x^61 + 46027*x^60 + 146118*x^59 + 427347*x^58 + 1155432*x^57 + 2912873*x^56 + 6875608*x^55 + 15281029*x^54 + 32094658*x^53 + 63945531*x^52 + 121210914*x^51 + 219194198*x^50 + 378998758*x^49 + 627863648*x^48 + 998282344*x^47 + 1525746624*x^46 + 2244502676*x^45 + 3181886869*x^44 + 4351201210*x^43 + 5744918381*x^42 + 7328807372*x^41 + 9039504349*x^40 + 10785767638*x^39 + 12455264802*x^38 + 13925287384*x^37 + 15077477135*x^36 + 15812782150*x^35 + 16065602576*x^34 + 15812782150*x^33 + 15077477135*x^32 + 13925287384*x^31 + 12455264802*x^30 + 10785767638*x^29 + 9039504349*x^28 + 7328807372*x^27 + 5744918381*x^26 + 4351201210*x^25 + 3181886869*x^24 + 2244502676*x^23 + 1525746624*x^22 + 998282344*x^21 + 627863648*x^20 + 378998758*x^19 + 219194198*x^18 + 121210914*x^17 + 63945531*x^16 + 32094658*x^15 + 15281029*x^14 + 6875608*x^13 + 2912873*x^12 + 1155432*x^11 + 427347*x^10 + 146118*x^9 + 46027*x^8 + 13186*x^7 + 3441*x^6 + 794*x^5 + 175*x^4 + 32*x^3 + 10*x^2 - 2*x + 1)/((x^6 - 1)^2*(x^4 + x^3 + x^2 + x + 1)^6*(x^3 - x^2 + x - 1)^6*(x^2 + x + 1)^6*(x + 1)^10*(x - 1)^23).
%F a(n) = n^30/(30!*5!) + O(n^29). - _Vaclav Kotesovec_, Aug 09 2022
%o (PARI) Vec(G(5, x)*(1 - x) + O(x^40)) \\ G defined in A028657. - _Andrew Howroyd_, Feb 28 2023
%Y Column 5 of A055080.
%Y First differences of A052264.
%Y Cf. A002620, A005783, A005784, A055066.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Jun 03 2000
%E a(0) = 1 prepended by _Stefano Spezia_, Aug 09 2022
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