A301736 Numerator of cumulative weight of certain D-forests on n nodes.

1, 0, 1, 1, 7, 11, 281, 449, 20719, 75403, 3066769, 1234967, 821856311, 2223747371, 273942958057, 1238828105761, 12489209350781, 511763293389419, 13479473195610647, 356089289643109313, 78908612931754624999, 373825489242185563339, 83933730864756536571961

Offset

0,5

Links

Table of n, a(n) for n=0..22.

Bernhard Gittenberger, Emma Yu Jin, Michael Wallner, On the shape of random Pólya structures, arXiv|1707.02144 [math.CO], 2017; Discrete Math., 341 (2018), 896-911.

Example

1, 0, 1/2, 1/3, 7/8, 11/30, 281/144, 449/840, ...

Mathematica

TreeGf[nn_] := Module[{A}, A = Table[1, nn]; For[n = 1, n <= nn-1, n++, A[[n+1]] = 1/n Sum[Sum[d A[[d]], {d, Divisors[k]}] A[[n-k+1]], {k, 1, n}] ]; A];
fracts[nn_] := Module[{v, t}, v = Table[0, nn+1]; t = TreeGf[nn]; v[[1]]=1; For[n=2, n <= nn, n++, v[[n+1]] = Sum[v[[n-i+1]] Sum[If[d != i, d t[[d]], 0], {d, Divisors[i]}], {i, 2, n}]/n]; v];
fracts[22] // Numerator (* Jean-François Alcover, Aug 05 2018, after Andrew Howroyd *)

Prog

(PARI) \\ See reference for recursion; TreeGf is gf of A000081.
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)}
fracts(n)={my(v=vector(n+1), t=Vec(TreeGf(n))); v[1]=1; for(n=2, n, v[n+1]=sum(i=2, n, v[n-i+1]*sumdiv(i, d, if(d<>i, d*t[d])))/n); v}
seq(n)={apply(f->numerator(f), fracts(n))} \\ Andrew Howroyd, Jun 21 2018

Crossrefs

Cf. A301737.

Sequence in context: A322950 A201120 A164328 * A096952 A143602 A177999

Adjacent sequences: A301733 A301734 A301735 * A301737 A301738 A301739

Keyword

nonn,frac

Author

N. J. A. Sloane, Apr 02 2018

Extensions

a(8)-a(22) from Andrew Howroyd, Jun 21 2018

Status

approved