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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
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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Last modified June 21 19:12 EDT 2021. Contains 345365 sequences. (Running on oeis4.)