cp's OEIS Frontend

A060053 Number of r-bicoverings (or restricted proper 2-covers) of an n-set.

%I A060053 #49 Aug 17 2024 03:25:40
%S A060053 1,0,1,5,43,518,8186,163356,3988342,116396952,3985947805,157783127673,
%T A060053 7131072006829,364166073164914,20827961078794845,1323968417981743817,
%U A060053 92917890994442697487,7157607311779373890120,602043767970637640566684
%N A060053 Number of r-bicoverings (or restricted proper 2-covers) of an n-set.
%C A060053 A bicovering is called an r-bicovering if the intersection of every two blocks contains at most one element.
%C A060053 Another name for this sequence is the number of restricted proper 2-covers of [1,...,n].
%C A060053 Number of T_0 2-regular set-systems on an n-set. - _Andrew Howroyd_, Jan 08 2020
%D A060053 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983. (See p. 203.)
%H A060053 Robert Gerbicz, <a href="/A060053/b060053.txt">Table of n, a(n) for n = 0..100</a>
%H A060053 Peter Cameron, Thomas Prellberg, and Dudley Stark, <a href="http://arxiv.org/abs/0707.0664">Asymptotic enumeration of 2-covers and line graphs</a>, arXiv:0707.0664 [math.CO], 2007.
%H A060053 Peter Cameron, Thomas Prellberg, and Dudley Stark, <a href="http://dx.doi.org/10.1016/j.disc.2008.09.008">Asymptotic enumeration of 2-covers and line graphs</a>, Discrete Math. 310 (2010), no. 2, 230-240. (See v_n.)
%F A060053 E.g.f. for number of k-block r-bicoverings of an n-set is exp(-x-x^2*y/2)*Sum_{i=0..inf} (1+y)^binomial(i, 2)*x^i/i!.
%F A060053 a(n) = row sums of A060052.
%F A060053 Inverse binomial transform of A014500. - _Vladeta Jovovic_, Aug 22 2006
%F A060053 The e.g.f.'s of A002718 (T(x)) and A060053 (V(x)) are related by T(x) = V(e^x-1).
%F A060053 The e.g.f.'s of A014500 (U(x)) and A060053 (V(x)) are related by U(x) = e^x*V(x).
%F A060053 E.g.f.: exp(-x/2)*(Sum_{k>=0} A020556(k)*(log(1 + x)/2)^k/k!). - _Andrew Howroyd_, Jan 13 2020
%e A060053 There are 5 r-bicoverings of a 3-set: 1 3-block bicovering {{1, 2}, {1, 3}, {2, 3}} and 4 4-block bicoverings {{1}, {2}, {3}, {1, 2, 3}}, {{2}, {3}, {1, 2}, {1, 3}}, {{1}, {3}, {1, 2}, {2, 3}}, {{1}, {2}, {1, 3}, {2, 3}}.
%e A060053 G.f. = 1 + x^2 + 5*x^3 + 43*x^4 + 518*x^5 + 8186*x^6 + 163356*x^7 + ...
%p A060053 A060053 := proc(n) local h, m; h := (m,n) -> add((-1/2)^k*binomial(m*(m-1)/2,n-k)/k!, k=0..n); n!*add(h(m,n)/m!, m=0..3*n); ceil(evalf(%/exp(1),99)) end: seq(A060053(i), i=0..18);
%p A060053 # Caveat computator! Limited accuracy. Do not use it for n > 50. - _Peter Luschny_, Jul 06 2011
%t A060053 f[n_] := FullSimplify[(n!/E)*Sum[(1/m!)*Sum[(-1/2)^k*Binomial[m*(m - 1)/2,
%t A060053 n - k]/k!, {k, 0, n}], {m, 0, Infinity}]] (* _Robert G. Wilson v_, Jul 03 2011 *)
%o A060053 (PARI) a(n)=round(n!/exp(1)*sum(m=0,3*n+1,1/m!*sum(k=0,n,(-1/2)^k*binomial(m*(m-1)/2,n-k)/k!)))
%o A060053 (PARI) \\ here egf1 is A020556 as e.g.f.
%o A060053 egf1(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, (-1)^k*binomial(i,k)*polcoef(bell, 2*i-k))*x^i/i!) + O(x*x^n)}
%o A060053 seq(n)={my(A=egf1(n), B=log(1+x + O(x*x^n))/2); Vec(serlaplace(exp(-x/2 + O(x*x^n))*sum(k=0, n, polcoef(A,k)*B^k)))} \\ _Andrew Howroyd_, Jan 13 2020
%Y A060053 Row 2 of A331039.
%Y A060053 Row sums of A060052.
%Y A060053 Cf. A060051, A002718, A059443, A003462, A059945-A059951.
%K A060053 easy,nonn
%O A060053 0,4
%A A060053 _Vladeta Jovovic_, Feb 15 2001