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

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, 62, 35, 14, 5, 1, 1, 286, 239, 169, 97, 46, 18, 5, 1, 1, 719, 622, 451, 282, 145, 63, 21, 6, 1, 1, 1842, 1607, 1217, 792, 440, 206, 80, 25, 6, 1, 1

Offset

1,4

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Formula

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.

Example

Triangle begins:
    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,  62,  35,  14,  5,  1, 1;
  286, 239, 169,  97,  46, 18,  5, 1, 1;
  719, 622, 451, 282, 145, 63, 21, 6, 1, 1;
  ...

Prog

(PARI) \\ TreeGf is A000081 as g.f.
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)}
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)}
M(n, m=n)=Mat(vector(m, k, ColSeq(n, k)~))
{ my(T=M(12)); for(n=1, #T~, print(T[n, 1..n])) }

Crossrefs

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

Row sums are A002861.

Cf. A033185, A217781, A339067.

Sequence in context: A112682 A033185 A217781 * A204849 A105632 A091491

Adjacent sequences: A339425 A339426 A339427 * A339429 A339430 A339431

Keyword

nonn,tabl

Author

Andrew Howroyd, Dec 03 2020

Status

approved