cp's OEIS Frontend

A127119 Triangle read by rows: T(n,k) = number of endofunctions on a set with n elements, where the maximum indegree is k.

%I A127119 #10 Feb 21 2020 17:05:05
%S A127119 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,
%T A127119 1,22,366,368,138,41,12,3,1,30,857,1092,439,140,41,12,3,1,42,2050,
%U A127119 3179,1395,455,140,41,12,3,1,56,4828,9160,4326,1467,457,140,41,12,3,1
%N A127119 Triangle read by rows: T(n,k) = number of endofunctions on a set with n elements, where the maximum indegree is k.
%C A127119 The number of endofunctions with indegree <= k is given by the Euler transform of the number of connected endofunctions with indegree <= k. - _Andrew Howroyd_, Feb 21 2020
%H A127119 Andrew Howroyd, <a href="/A127119/b127119.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%e A127119 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.
%e A127119 The triangle starts:
%e A127119 1
%e A127119 2 1
%e A127119 3 3 1
%e A127119 5 10 3 1
%e A127119 7 24 12 3 1
%o A127119 (PARI) \\ Here R(n,k) gives column k of A299038 as series.
%o A127119 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A127119 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)}
%o A127119 CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
%o A127119 R(n,k)={my(p=O(x)); for(n=1, n, p=x*MSetUptoK(p, k)); p}
%o A127119 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]])}
%o A127119 { my(A=F(12)); for(n=1, #A, print(A[n, 1..n])) } \\ _Andrew Howroyd_, Feb 21 2020
%Y A127119 Cf. A127120, A299038.
%K A127119 nonn,tabl
%O A127119 1,2
%A A127119 _Franklin T. Adams-Watters_, Jan 05 2007
%E A127119 Terms a(46) and beyond from _Andrew Howroyd_, Feb 21 2020