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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-François 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 EXTENSIONS a(0)=1 prepended by Andrew Howroyd, Sep 09 2018 STATUS approved

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Last modified April 11 14:58 EDT 2021. Contains 342886 sequences. (Running on oeis4.)