A127119 Triangle read by rows: T(n,k) = number of endofunctions on a set with n elements, where the maximum indegree is k.
1, 2, 1, 3, 3, 1, 5, 10, 3, 1, 7, 24, 12, 3, 1, 11, 64, 39, 12, 3, 1, 15, 149, 122, 41, 12, 3, 1, 22, 366, 368, 138, 41, 12, 3, 1, 30, 857, 1092, 439, 140, 41, 12, 3, 1, 42, 2050, 3179, 1395, 455, 140, 41, 12, 3, 1, 56, 4828, 9160, 4326, 1467, 457, 140, 41, 12, 3, 1
Offset: 1
Examples
For n = 3, the 7 endofunctions are (1,2,3) -> (1,1,1), (1,1,2), (1,2,1), (2,1,1), (1,2,3), (1,3,2) and (2,3,1). In the first, node 1 has indegree 3, the next 3 have node 1 with indegree 2 and the final 3 are permutations, each node having indegree 1. So row 3 of the triangle is 3,3,1. The triangle starts: 1 2 1 3 3 1 5 10 3 1 7 24 12 3 1
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Programs
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PARI
\\ Here R(n,k) gives column k of A299038 as series. EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)} MSetUptoK(g, k)={my(n=serprec(g,x)); polcoef(if(k==0, 1, exp( sum(i=1, k, (y^i + O(y*y^k))*subst(g + O(x*x^(n\i)), x, x^i)/i )))/(1 - y) + O(y*y^k), k, y) + O(x^n)} CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))} R(n,k)={my(p=O(x)); for(n=1, n, p=x*MSetUptoK(p, k)); p} F(n)={my(M=Mat(vector(n, k, EulerT(Vec(CIK(x*MSetUptoK(R(n,k), k-1), n)))~))); M-matconcat([vectorv(#M), M[, 1..n-1]])} { my(A=F(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Feb 21 2020
Extensions
Terms a(46) and beyond from Andrew Howroyd, Feb 21 2020
Comments