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%I A088309 #21 Dec 17 2022 08:26:03
%S A088309 1,2,5,44,1411,159656,62055868,82060884560,371036717493194,
%T A088309 5812014504668066528,320454239459072905856944,
%U A088309 63156145369562679089674952768,45090502574837184532027563736271152,117910805393665959622047902193019284914432,1139353529410754170844431642119963019965901238144
%N A088309 Number of equivalence classes of n X n (0,1)-matrices with all rows distinct and all columns distinct.
%C A088309 Two such matrices are equivalent if they differ just by a permutation of the rows.
%H A088309 G. C. Greubel, <a href="/A088309/b088309.txt">Table of n, a(n) for n = 0..59</a>
%H A088309 G. Kilibarda and V. Jovovic, <a href="https://arxiv.org/abs/1411.4187">Enumeration of some classes of T_0-hypergraphs</a>, arXiv:1411.4187 [math.CO], 2014.
%F A088309 a(n) = Sum_{k=0..n} Stirling1(n, k)*binomial(2^k, n). - _Vladeta Jovovic_, Nov 07 2003
%F A088309 a(n) = A088310(n) / n!.
%e A088309 a(2) = 5: 00/01, 00/10, 01/10, 01/11, 10/11.
%t A088309 A088309[n_]:= A088309[n]=Sum[Binomial[2^j,n]*StirlingS1[n,j], {j,0,n}];
%t A088309 Table[A088309[n], {n,0,30}] (* _G. C. Greubel_, Dec 15 2022 *)
%o A088309 (Magma)
%o A088309 A088309:= func< n | (&+[Binomial(2^k,n)*StirlingFirst(n,k): k in [0..n]]) >;
%o A088309 [A088309(n): n in [0..30]]; // _G. C. Greubel_, Dec 15 2022
%o A088309 (SageMath)
%o A088309 @CachedFunction
%o A088309 def A088309(n): return (-1)^n*sum((-1)^k*binomial(2^k, n)*stirling_number1(n, k) for k in (0..n))
%o A088309 [A088309(n) for n in range(31)] # _G. C. Greubel_, Dec 15 2022
%o A088309 (PARI) a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(2^k, n)); \\ _Michel Marcus_, Dec 16 2022
%Y A088309 Cf. A088229, A088310, A088616.
%Y A088309 Main diagonal of A059084.
%Y A088309 Binary matrices with distinct rows and columns, various versions: A059202, this sequence, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763.
%K A088309 nonn
%O A088309 0,2
%A A088309 _N. J. A. Sloane_, Nov 07 2003
%E A088309 Suggested by Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 06 2003
%E A088309 a(0)-a(5) from _W. Edwin Clark_, Nov 07 2003

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