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%I #30 Oct 04 2016 13:16:06
%S 1,4,4,13,39,25,3,40,280,472,256,40,121,1815,6185,7255,3306,535,15,
%T 364,11284,70700,149660,131876,51640,8456,420,1093,68859,759045,
%U 2681063,3961356,2771685,954213,154637,9730,105,3280,416560,7894992,44659776,103290096
%N Triangle T(n,k) (n >= 2, k = 3..n+floor(n/2)) giving number of bicoverings of an n-set with k blocks.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #40.
%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
%H Alois P. Heinz, <a href="/A059443/b059443.txt">Rows n = 2..60, flattened</a>
%H 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 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 T(n, k) = Sum{j=0..n} Stirling2(n, j) * A060052(j, k). - _David Pasino_, Sep 22 2016
%e T(2,3) = 1: 1|12|2.
%e T(3,3) = 4: 1|123|23, 12|13|23, 12|123|3, 123|13|2.
%e T(3,4) = 4: 1|12|23|3, 1|13|2|23, 1|123|2|3, 12|13|2|3.
%e Triangle T(n,k) begins:
%e : 1;
%e : 4, 4;
%e : 13, 39, 25, 3;
%e : 40, 280, 472, 256, 40;
%e : 121, 1815, 6185, 7255, 3306, 535, 15;
%e : 364, 11284, 70700, 149660, 131876, 51640, 8456, 420;
%e : 1093, 68859, 759045, 2681063, 3961356, 2771685, 954213, 154637, 9730, 105;
%e ...
%t 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 (PARI) \ps 22;
%o s = 8; pv = vector(s); for(n=1,s,pv[n]=round(polcoeff(f(x,y),n,y)*n!));
%o for(n=1,s,for(m=3,poldegree(pv[n],x),print1(polcoeff(pv[n],m),", "))) \\ _Gerald McGarvey_, Dec 03 2009
%Y Columns k=3-10 give: A003462, A059945, A059946, A059947, A059948, A059949, A059950, A059951.
%Y Row sums are A002718.
%Y Main diagonal gives A275517.
%Y Right border gives A275521.
%K tabf,nonn,nice
%O 2,2
%A _N. J. A. Sloane_, Feb 01 2001
%E More terms and additional comments from _Vladeta Jovovic_, Feb 14 2001
%E a(37) corrected by _Gerald McGarvey_, Dec 03 2009
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