A060052 - OEIS

%I #22 Jan 30 2020 18:14:09

%S 1,1,4,0,15,25,3,0,30,222,226,40,0,30,1230,3670,2706,535,15,0,0,5040,

%T 39900,69450,40405,8141,420,0,0,15120,345240,1254960,1498035,722275,

%U 142877,9730,105,0,0,30240,2492280,18587520,40701780,36450820,15031204,2871240,226828,5040

%N Triangle read by rows: T(n,k) gives number of r-bicoverings of an n-set with k blocks, n >= 2, k = 3..n+floor(n/2).

%C A bicovering is r-bicovering if intersection of every two blocks contains at most one element.

%D I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

%H Andrew Howroyd, <a href="/A060052/b060052.txt">Table of n, a(n) for n = 2..1802</a> (rows n=2..50)

%H <a href="/A060052/a060052.pdf">Table</a>

%F E.g.f.: A(x, y) = exp(-x-1/2*x^2*y)*Sum_{i>=0} (1+y)^binomial(i, 2)*x^i/i!.

%F T(n, k) = (n!/k!) * A276640(k, n). - _David Pasino_, Sep 22 2016

%F T(n,k) = 0 for n > binomial(k,2). - _Andrew Howroyd_, Jan 30 2020

%e Triangle starts:

%e [1],

%e [1, 4],

%e [0, 15, 25, 3],

%e [0, 30, 222, 226, 40],

%e [0, 30, 1230, 3670, 2706, 535, 15],

%e [0, 0, 5040, 39900, 69450, 40405, 8141, 420],

%e [0, 0, 15120, 345240, 1254960, 1498035, 722275, 142877, 9730, 105],

%e [0, 0, 30240, 2492280, 18587520, 40701780, 36450820, 15031204, 2871240, 226828, 5040],

%e ...

%o (PARI) \\ returns k-th column as vector.

%o C(k)=if(k<3, [], Vecrev(serlaplace(polcoef(exp(-x-1/2*x^2*y + O(x*x^k))*sum(i=0, 3*k\2, (1+y)^binomial(i, 2)*x^i/i!), k))/y)) \\ _Andrew Howroyd_, Jan 30 2020

%o (PARI)

%o T(n)={my(m=(3*n\2), y='y + O('y^(n+1))); my(g=exp(-x-1/2*x^2*y + O(x*x^m))*sum(k=0, m, (1+y)^binomial(k, 2)*x^k/k!)); Mat([Col(serlaplace(p), -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 A060053.

%Y Column sums are A060051.

%Y Cf. A059443, A059530, A060092, A060487, A060492, A276640, A331039.

%K easy,nonn,tabf

%O 2,3

%A _Vladeta Jovovic_, Feb 15 2001

%E Zeros inserted into data by _Andrew Howroyd_, Jan 30 2020