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A059443 Triangle T(n,k) (n >= 2, k = 3..n+floor(n/2)) giving number of bicoverings of an n-set with k blocks.

%I A059443 #34 Apr 18 2025 05:49:09
%S A059443 1,4,4,13,39,25,3,40,280,472,256,40,121,1815,6185,7255,3306,535,15,
%T A059443 364,11284,70700,149660,131876,51640,8456,420,1093,68859,759045,
%U A059443 2681063,3961356,2771685,954213,154637,9730,105,3280,416560,7894992,44659776,103290096
%N A059443 Triangle T(n,k) (n >= 2, k = 3..n+floor(n/2)) giving number of bicoverings of an n-set with k blocks.
%D A059443 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #40.
%D A059443 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
%H A059443 Alois P. Heinz, <a href="/A059443/b059443.txt">Rows n = 2..60, flattened</a>
%H A059443 L. Comtet, <a href="/A002718/a002718.pdf">Birecouvrements et birevêtements d'un ensemble fini</a>, Studia Sci. Math. Hungar 3 (1968): 137-152. [Annotated scanned copy. Warning: the table of v(n,k) has errors.]
%F A059443 E.g.f. for m-block bicoverings of an n-set is exp(-x-1/2*x^2*(exp(y)-1))*Sum_{i=0..inf} x^i/i!*exp(binomial(i, 2)*y).
%F A059443 T(n, k) = Sum{j=0..n} Stirling2(n, j) * A060052(j, k). - _David Pasino_, Sep 22 2016
%e A059443 T(2,3) = 1: 1|12|2.
%e A059443 T(3,3) = 4: 1|123|23, 12|13|23, 12|123|3, 123|13|2.
%e A059443 T(3,4) = 4: 1|12|23|3, 1|13|2|23, 1|123|2|3, 12|13|2|3.
%e A059443 Triangle T(n,k) begins:
%e A059443 :    1;
%e A059443 :    4,     4;
%e A059443 :   13,    39,     25,       3;
%e A059443 :   40,   280,    472,     256,      40;
%e A059443 :  121,  1815,   6185,    7255,    3306,     535,     15;
%e A059443 :  364, 11284,  70700,  149660,  131876,   51640,   8456,    420;
%e A059443 : 1093, 68859, 759045, 2681063, 3961356, 2771685, 954213, 154637, 9730, 105;
%e A059443 ...
%t A059443 nmax = 8; imax = 2*(nmax - 2); egf := E^(-x - 1/2*x^2*(E^y - 1))*Sum[(x^i/i!)*E^(Binomial[i, 2]*y), {i, 0, imax}]; fx = CoefficientList[ Series[ egf , {y, 0, imax}], y]*Range[0, imax]!; row[n_] := Drop[ CoefficientList[ Series[fx[[n + 1]], {x, 0, imax}], x], 3]; Table[ row[n], {n, 2, nmax}] // Flatten (* _Jean-François Alcover_, Sep 21 2012 *)
%o A059443 (PARI) \ps 22;
%o A059443 s = 8; pv = vector(s); for(n=1,s,pv[n]=round(polcoeff(f(x,y),n,y)*n!));
%o A059443 for(n=1,s,for(m=3,poldegree(pv[n],x),print1(polcoeff(pv[n],m),", "))) \\ _Gerald McGarvey_, Dec 03 2009
%Y A059443 Columns k=3-10 give: A003462, A059945, A059946, A059947, A059948, A059949, A059950, A059951.
%Y A059443 Row sums are A002718.
%Y A059443 Main diagonal gives A275517.
%Y A059443 Right border gives A275521.
%K A059443 tabf,nonn,nice
%O A059443 2,2
%A A059443 _N. J. A. Sloane_, Feb 01 2001
%E A059443 More terms and additional comments from _Vladeta Jovovic_, Feb 14 2001
%E A059443 a(37) corrected by _Gerald McGarvey_, Dec 03 2009