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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. 15

%I

%S 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 62,35,14,5,1,1,286,239,169,97,46,18,5,1,1,719,622,451,282,145,63,21,

%U 6,1,1,1842,1607,1217,792,440,206,80,25,6,1,1

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

%H Andrew Howroyd, <a href="/A339428/b339428.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)

%F 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 Triangle begins:

%e 1;

%e 1, 1;

%e 2, 1, 1;

%e 4, 3, 1, 1;

%e 9, 6, 3, 1, 1;

%e 20, 16, 9, 4, 1, 1;

%e 48, 37, 23, 11, 4, 1, 1;

%e 115, 96, 62, 35, 14, 5, 1, 1;

%e 286, 239, 169, 97, 46, 18, 5, 1, 1;

%e 719, 622, 451, 282, 145, 63, 21, 6, 1, 1;

%e ...

%o (PARI) \\ TreeGf is A000081 as g.f.

%o 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 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 M(n, m=n)=Mat(vector(m, k, ColSeq(n,k)~))

%o { my(T=M(12)); for(n=1, #T~, print(T[n,1..n])) }

%Y Columns 1..12 are A000081, A027852, A029852, A029853, A029868, A029869, A029870, A029871, A032205, A032206, A032207, A032208.

%Y Row sums are A002861.

%Y Cf. A033185, A217781, A339067.

%K nonn,tabl

%O 1,4

%A _Andrew Howroyd_, Dec 03 2020

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Last modified July 24 04:25 EDT 2021. Contains 346273 sequences. (Running on oeis4.)