A059946 -id:A059946 - cp's OEIS frontend

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

1, 4, 4, 13, 39, 25, 3, 40, 280, 472, 256, 40, 121, 1815, 6185, 7255, 3306, 535, 15, 364, 11284, 70700, 149660, 131876, 51640, 8456, 420, 1093, 68859, 759045, 2681063, 3961356, 2771685, 954213, 154637, 9730, 105, 3280, 416560, 7894992, 44659776, 103290096

Offset: 2

N. J. A. Sloane, Feb 01 2001

			T(2,3) = 1: 1|12|2.
T(3,3) = 4: 1|123|23, 12|13|23, 12|123|3, 123|13|2.
T(3,4) = 4: 1|12|23|3, 1|13|2|23, 1|123|2|3, 12|13|2|3.
Triangle T(n,k) begins:
:    1;
:    4,     4;
:   13,    39,     25,       3;
:   40,   280,    472,     256,      40;
:  121,  1815,   6185,    7255,    3306,     535,     15;
:  364, 11284,  70700,  149660,  131876,   51640,   8456,    420;
: 1093, 68859, 759045, 2681063, 3961356, 2771685, 954213, 154637, 9730, 105;
...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #40.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Alois P. Heinz, Rows n = 2..60, flattened
L. Comtet, Birecouvrements et birevêtements d'un ensemble fini, Studia Sci. Math. Hungar 3 (1968): 137-152. [Annotated scanned copy. Warning: the table of v(n,k) has errors.]

Columns k=3-10 give: A003462, A059945, A059946, A059947, A059948, A059949, A059950, A059951.
Row sums are A002718.
Main diagonal gives A275517.
Right border gives A275521.

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 *)

\ps 22;
s = 8; pv = vector(s); for(n=1,s,pv[n]=round(polcoeff(f(x,y),n,y)*n!));
for(n=1,s,for(m=3,poldegree(pv[n],x),print1(polcoeff(pv[n],m),", "))) \\ Gerald McGarvey, Dec 03 2009

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).

T(n, k) = Sum{j=0..n} Stirling2(n, j) * A060052(j, k). - David Pasino, Sep 22 2016

More terms and additional comments from Vladeta Jovovic, Feb 14 2001

a(37) corrected by Gerald McGarvey, Dec 03 2009

A060093 Number of 5-block ordered bicoverings of an unlabeled n-set.

Original entry on oeis.org

0, 0, 0, 0, 125, 722, 2565, 7180, 17335, 37750, 76093, 144340, 260590, 451440, 755040, 1224964, 1935050, 2985380, 4509590, 6683720, 9736835, 13963670, 19739575, 27538060, 37951265, 51713706, 69729675, 93104700, 123181500

Offset: 0

Vladeta Jovovic, Feb 26 2001

Harry J. Smith, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Column k=5 of A060092.

Cf. A059946, A060090, A060489.

a(n) = if(n<1, 0, binomial(n + 9, 9) - 5*binomial(n + 5, 5) - 10*binomial(n + 3, 3) + 20*binomial(n + 2, 2) + 30*binomial(n + 1, 1) - 60*binomial(n, 0) + 24*binomial(n - 1, -1)) \\ Harry J. Smith, Jul 01 2009

a(n) = binomial(n+9, 9) - 5*binomial(n+5, 5) - 10*binomial(n+3, 3) + 20*binomial(n+2, 2) + 30*binomial(n+1, 1) - 60*binomial(n, 0) + 24*binomial(n-1, -1).
G.f.: y^4*(-528*y + 125 + 970*y^2 - 980*y^3 + 570*y^4 - 180*y^5 + 24*y^6)/(-1 + y)^10.
E.g.f. for k-block ordered bicoverings of an unlabeled n-set is exp(-x - x^2/2*y/(1 - y))*Sum_{k>=0} 1/(1 - y)^binomial(k, 2)*x^k/k!.
a(n) = (n-1) *(n-2) *(n-3) *(n^6 + 51*n^5 + 1165*n^4 + 15885*n^3 + 130954*n^2 + 660504*n + 1451520)/ 362880, n > 0. - R. J. Mathar, Aug 10 2017

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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.

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