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A368951
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Number of connected labeled graphs with n edges and n vertices and with loops allowed.
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13
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1, 1, 2, 10, 79, 847, 11436, 185944, 3533720, 76826061, 1880107840, 51139278646, 1530376944768, 49965900317755, 1767387701671424, 67325805434672100, 2747849045156064256, 119626103584870552921, 5533218319763109888000, 270982462739224265922466
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OFFSET
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0,3
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COMMENTS
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LINKS
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Eric Weisstein's World of Mathematics, Graph Loop.
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FORMULA
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E.g.f.: 1 - log(1-T(x))/2 + T(x)/2 - T(x)^2/4 where T(x) = -LambertW(-x) is the e.g.f. of A000169.
a(n) = (exp(n)*Gamma(n + 1, n) - (n - 1)*n^(n - 1))/(2*n) for n > 0.
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EXAMPLE
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The a(0) = 1 through a(3) = 10 loop-graphs:
{} {11} {11,12} {11,12,13}
{22,12} {11,12,23}
{11,13,23}
{22,12,13}
{22,12,23}
{22,13,23}
{33,12,13}
{33,12,23}
{33,13,23}
{12,13,23}
(End)
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MAPLE
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egf:= (L-> 1-L/2-log(1+L)/2-L^2/4)(LambertW(-x)):
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
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PROG
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(PARI) seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(-log(1-t)/2 + t/2 - t^2/4 + 1))}
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CROSSREFS
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This is the connected covering case of A014068.
Allowing any number of edges gives A062740, connected case of A322661.
This is the connected case of A368597.
For at most n edges we have A369197.
A000085 counts set partitions into singletons or pairs.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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