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A006023 Number of unlabeled mating digraphs with n nodes.
(Formerly M2046)
6
1, 1, 2, 12, 183, 8884, 1495984, 872987584, 1787227218134, 13013640978954744, 341143259362180445672, 32519497484526664873838560, 11366387701006542223325518765872, 14668949294272099348849331250968826816 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

R. C. Read, The Enumeration of Mating-Type Graphs. Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, Oct 1989.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..50

R. C. Read, The Enumeration of Mating-Type Graphs, Dept. Combinatorics and Optimization, Univ. Waterloo, Oct 1989. (Annotated scanned copy)

FORMULA

G.f. Sum_{n>=1} x^n * (Sum_{(j)} h((j)) * 2^Y((j)) * Product_{i=1..n} (1-x^i)^{j_i}) where the inner sum runs over all partitions (j) = j_1j_2...j_m of n = 1 * j_1 + 2 * j_2 + ... _ m * j_m, h((j)) = 1 / Product_{i=1..m} (j_i! * i^{j_i}), and  Y((j)) = Sum_{i=1..m} (i-1) * j_i + Sum_{i=1..m} i * j_i * (j_i - 1) + 2 * Sum_{1 <= i < k <= m} j_i * j_k * gcd(i, k). - Sean A. Irvine, Mar 06 2018

MATHEMATICA

permcount[v_] := Module[{m = 1, s = 0, t, i, k = 0}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];

edges[v_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Sum[v[[i]] - 1, {i, 1, Length[v]}];

a[n_] := Module[{s = 0}, If[n == 0, Return[1]]; Sum[Do[ s += permcount[p]* 2^edges[p] * Coefficient[Product[1 - x^p[[i]], {i, 1, Length[p]}], x, n - k]/k!, {p, IntegerPartitions[k]}], {k, 1, n}]; s];

a /@ Range[0, 20] (* Jean-Fran├žois Alcover, Sep 22 2019, after Andrew Howroyd *)

PROG

(PARI) \\ compare A000273.

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

edges(v) = {sum(i=2, #v, sum(j=1, i-1, 2*gcd(v[i], v[j]))) + sum(i=1, #v, v[i]-1)}

a(n) = {if(n<1, n==0, my(s=0); sum(k=1, n, forpart(p=k, s+=permcount(p) * 2^edges(p) * polcoef(prod(i=1, #p, 1-x^p[i]), n-k)/k!)); s)} \\ Andrew Howroyd, Sep 09 2018

CROSSREFS

Cf. A000273, A004110, A006025.

Sequence in context: A067962 A134716 A243807 * A039748 A007764 A015195

Adjacent sequences:  A006020 A006021 A006022 * A006024 A006025 A006026

KEYWORD

nonn

AUTHOR

Simon Plouffe

EXTENSIONS

a(0)=1 prepended by Andrew Howroyd, Sep 09 2018

STATUS

approved

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Last modified February 24 09:40 EST 2020. Contains 332209 sequences. (Running on oeis4.)