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A048194
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Total number of split graphs (chordal + chordal complement) on n vertices.
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16
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1, 2, 4, 9, 21, 56, 164, 557, 2223, 10766, 64956, 501696, 5067146, 67997750, 1224275498, 29733449510, 976520265678, 43425320764422, 2616632636247976, 213796933371366930, 23704270652844196754, 3569464106212250952762, 730647291666881838671052
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OFFSET
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1,2
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COMMENTS
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Also number of bipartite graphs with n vertices and no isolated vertices in distinguished bipartite block, up to isomorphism; so a(n) equals first differences of A049312. - Vladeta Jovovic, Jun 17 2000
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LINKS
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FORMULA
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a(n) ~ A049312(n) ~ (1/n!) * Sum_{k=0..n} binomial(n,k) * 2^(k(n-k)) (see the Troyka link, Thms. 3.7 and 3.10). - Justin M. Troyka, Oct 29 2018
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[ Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
a[d_] := Sum[A[n, d - n], {n, 0, d}] - Sum[A[n, d - n - 1], {n, 0, d - 1}];
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CROSSREFS
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Detlef Pauly remarks that this is the unlabeled analog of A001831.
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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