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A058863
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Number of connected labeled chordal graphs on n nodes with no induced path P_4; also the number of labeled trees with each vertex replaced by a clique.
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6
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1, 1, 4, 23, 181, 1812, 22037, 315569, 5201602, 97009833, 2019669961, 46432870222, 1168383075471, 31939474693297, 942565598033196, 29866348653695203, 1011335905644178273, 36446897413531401020, 1392821757824071815641, 56259101478392975833333
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OFFSET
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1,3
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COMMENTS
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A subclass of chordal-comparability graphs.
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LINKS
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FORMULA
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1) c(n) = 1 + Sum_{k=1..n-2} binomial(n, k)*(t(n-k) - c(n-k))
2) t(n) = c(n) + Sum_{k=1..n-1} k*c(k)*binomial(n, k)*t(n-k))/n
where c(n) (A058863) is the number of connected graphs of this type and t(n) (A058864) is the total number of such graphs.
a(n) is asymptotic to sqrt(r*(e-1))/n*(n/(e*r))^n where r = 1 - log(e-1).
a(n) = Sum_{k=0..n} Stirling2(n, k)*A060356(k). Also a(n) = Sum_{k=1..n} (-1)^(n-k)*Stirling2(n, k)*k^(k-1). - Vladeta Jovovic, Sep 17 2003
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MAPLE
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S:= series(-LambertW(exp(-x)-1), x, 101):
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MATHEMATICA
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a[n_] := Sum[(-1)^(n-k)*StirlingS2[n, k]*k^(k-1), {k, 1, n}];
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PROG
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(PARI)
geta(n, va, vA) = {local(k); if (n==1, return(1)); if (n==2, return(1)); return(1 + sum(k=1, n-2, binomial(n, k)*(vA[n-k] - va[n-k]))); }
getA(n, va, vA) = {local(k); if (n==1, return(1)); if (n==2, return(2)); return ((va[n] + sum(k=1, n-1, k*va[k]*binomial(n, k)*vA[n-k])/n)); }
both(n) = {va = vector(n); vA = vector(n); for (i=1, n, va[i] = geta(i, va, vA); vA[i] = getA(i, va, vA); ); print("va_A058863=", va); print("vA_A058864=", vA); }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Robert Castelo (rcastelo(AT)imim.es), Jan 06 2001
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EXTENSIONS
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STATUS
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approved
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