A060052 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).
1, 1, 4, 0, 15, 25, 3, 0, 30, 222, 226, 40, 0, 30, 1230, 3670, 2706, 535, 15, 0, 0, 5040, 39900, 69450, 40405, 8141, 420, 0, 0, 15120, 345240, 1254960, 1498035, 722275, 142877, 9730, 105, 0, 0, 30240, 2492280, 18587520, 40701780, 36450820, 15031204, 2871240, 226828, 5040
Offset: 2
Examples
Triangle starts: [1], [1, 4], [0, 15, 25, 3], [0, 30, 222, 226, 40], [0, 30, 1230, 3670, 2706, 535, 15], [0, 0, 5040, 39900, 69450, 40405, 8141, 420], [0, 0, 15120, 345240, 1254960, 1498035, 722275, 142877, 9730, 105], [0, 0, 30240, 2492280, 18587520, 40701780, 36450820, 15031204, 2871240, 226828, 5040], ...
References
- I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
Links
- Andrew Howroyd, Table of n, a(n) for n = 2..1802 (rows n=2..50)
- Table
Crossrefs
Programs
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PARI
\\ returns k-th column as vector. 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
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PARI
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]])} { my(A=T(8)); for(n=2, matsize(A)[1], print(A[n, 3..3*n\2])) } \\ Andrew Howroyd, Jan 30 2020
Formula
E.g.f.: A(x, y) = exp(-x-1/2*x^2*y)*Sum_{i>=0} (1+y)^binomial(i, 2)*x^i/i!.
T(n, k) = (n!/k!) * A276640(k, n). - David Pasino, Sep 22 2016
T(n,k) = 0 for n > binomial(k,2). - Andrew Howroyd, Jan 30 2020
Extensions
Zeros inserted into data by Andrew Howroyd, Jan 30 2020
Comments