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A339428 Triangle read by rows: T(n,k) is the number of connected functions on n points with a loop of length k.

%I A339428 #10 Dec 08 2020 15:24:39
%S A339428 1,1,1,2,1,1,4,3,1,1,9,6,3,1,1,20,16,9,4,1,1,48,37,23,11,4,1,1,115,96,
%T A339428 62,35,14,5,1,1,286,239,169,97,46,18,5,1,1,719,622,451,282,145,63,21,
%U A339428 6,1,1,1842,1607,1217,792,440,206,80,25,6,1,1
%N A339428 Triangle read by rows: T(n,k) is the number of connected functions on n points with a loop of length k.
%H A339428 Andrew Howroyd, <a href="/A339428/b339428.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)
%F A339428 G.f. of k-th column: (1/k)*Sum_{d|k} phi(d) * r(x^d)^(k/d) where r(x) is the g.f. of A000081.
%e A339428 Triangle begins:
%e A339428     1;
%e A339428     1,   1;
%e A339428     2,   1,   1;
%e A339428     4,   3,   1,   1;
%e A339428     9,   6,   3,   1,   1;
%e A339428    20,  16,   9,   4,   1,  1;
%e A339428    48,  37,  23,  11,   4,  1,  1;
%e A339428   115,  96,  62,  35,  14,  5,  1, 1;
%e A339428   286, 239, 169,  97,  46, 18,  5, 1, 1;
%e A339428   719, 622, 451, 282, 145, 63, 21, 6, 1, 1;
%e A339428   ...
%o A339428 (PARI) \\ TreeGf is A000081 as g.f.
%o A339428 TreeGf(N) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
%o A339428 ColSeq(n,k)={my(r=TreeGf(max(0,n+1-k))); Vec(sumdiv(k, d, eulerphi(d)*subst(r + O(x*x^(n\d)), x, x^d)^(k/d))/k, -n)}
%o A339428 M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))
%o A339428 { my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }
%Y A339428 Columns 1..12 are A000081, A027852, A029852, A029853, A029868, A029869, A029870, A029871, A032205, A032206, A032207, A032208.
%Y A339428 Row sums are A002861.
%Y A339428 Cf. A033185, A217781, A339067.
%K A339428 nonn,tabl
%O A339428 1,4
%A A339428 _Andrew Howroyd_, Dec 03 2020