%I #18 Jan 30 2020 14:21:45

%S 3,7,16,12,63,125,90,18,162,722,1716,1680,25,341,2565,11350,27342,

%T 29960,7560,33,636,7180,49860,208302,503000,631512,302400,42,1092,

%U 17335,173745,1099602,4389875,10762299,14975730,9632700,1247400

%N Triangle T(n,k) of k-block ordered bicoverings of an unlabeled n-set, n >= 2, k = 3..n+floor(n/2).

%C All columns are polynomials of order binomial(k, 2). - _Andrew Howroyd_, Jan 30 2020

%H Andrew Howroyd, <a href="/A060092/b060092.txt">Table of n, a(n) for n = 2..1802</a>

%F 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..inf} 1/(1-y)^binomial(k, 2)*x^k/k!.

%e [3],

%e [7, 16],

%e [12, 63, 125, 90],

%e [18, 162, 722, 1716, 1680],

%e [25, 341, 2565, 11350, 27342, 29960, 7560],

%e [33, 636, 7180, 49860, 208302, 503000, 631512, 302400],

%e [42, 1092, 17335, 173745, 1099602, 4389875, 10762299, 14975730, 9632700, 1247400], ...

%e There are 23=7+16 ordered bicoverings of an unlabeled 3-set: 7 3-block bicoverings and 16 4-block bicoverings, cf. A060090.

%o (PARI) \\ gives g.f. of k-th column.

%o ColGf(k) = k!*polcoef(exp(-x - x^2*y/(2*(1-y)) + O(x*x^k))*sum(j=0, k, 1/(1-y)^binomial(j, 2)*x^j/j!), k) \\ _Andrew Howroyd_, Jan 30 2020

%o (PARI)

%o T(n)={my(m=(3*n\2), y='y + O('y^(n+1))); my(g=serlaplace(exp(-x - x^2*y/(2*(1-y)) + O(x*x^m))*sum(k=0, m, 1/(1-y)^binomial(k, 2)*x^k/k!))); Mat([Col(p/y^2, -n) | p<-Vec(g)[2..m+1]])}

%o { my(A=T(8)); for(n=2, matsize(A)[1], print(A[n, 3..3*n\2])) } \\ _Andrew Howroyd_, Jan 30 2020

%Y Row sums are A060090.

%Y Columns k=3..7 are A055998(n-1), A060091, A060093, A060094, A060095.

%Y Cf. A059443, A060052, A060487, A060492, A094573, A331571.

%K nonn,tabf

%O 2,1

%A _Vladeta Jovovic_, Feb 26 2001