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A308700
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a(n) = n * 2^(n - 2) * (2^(n - 1) - 1).
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0
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0, 0, 2, 18, 112, 600, 2976, 14112, 65024, 293760, 1308160, 5761536, 25153536, 109025280, 469704704, 2013143040, 8589672448, 36506664960, 154617643008, 652832538624, 2748773826560, 11544861081600, 48378488553472, 202310091276288, 844424829468672, 3518436999168000
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OFFSET
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0,3
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COMMENTS
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Given a pseudo-graph P of the set X = {1, 2, ..., n}, defined as a graph represented by the discrete topology on the set X (the power set of X), for n > 0, a(n) is the number of edges of the topological graph arising by deleting loops in P (see Theorem 3.3 in Kozae et al.).
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LINKS
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FORMULA
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O.g.f.: -2 * x^2 * (-1 + 3*x)/((-1 + 2*x)^2 * (-1 + 4*x)^2).
E.g.f.: (1/2) * exp(2*x) * (-1 + exp(2*x)) * x.
a(n) = 12 * a(n - 1) - 52*a(n - 2) + 96*a(n - 3) - 64*a(n - 4) for n > 3.
a(n) = n * 2^(n - 2) * (2^(n - 1) - 1).
Lim_{n -> infinity} a(n)/a(n - 1) = 4.
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EXAMPLE
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For n = 3, the set X = {1,2,3},
the power set 2^X = {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, X} and the pseudo-graph P represented by 2^X has the following edges, here grouped into...
simple loops:
{1} --- {1}, {2} --- {2}, {3} --- {3} for a total of 3.
double loops:
{1,2} --- {1,2}, {1,3} --- {1,3}, {2,3} --- {2,3} for a total of 6 simple loops.
triple loop:
X --- X for a total of 3 simple loops.
simple edges:
{1} --- {1,2}, {1} --- {1,3}, {1} --- X, {2} --- {1,2}, {2} --- {2,3}, {2} --- X, {3} --- {1,3}, {3} --- {2,3}, {3} --- X, {1,2} --- {1,3}, {1,2} --- {2,3}, {1,3} --- {2,3} for a total of 12.
double edges:
{1,2} --- X, {1,3} --- X, {2,3} --- X for a total of 6 simple edges.
By deleting the loops in P, there remain a total of a(3) = 12 + 6 = 18 edges for the topological graph arising from P.
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MAPLE
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a:=n->n*2^(n-2)*(2^(n-1)-1): seq(a(n), n=0..25);
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MATHEMATICA
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Table[n 2^(n - 2)(2^(n - 1) - 1), {n, 0, 31}]
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PROG
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(GAP) Flat(List([0..25], n->n*2^(n-2)*(2^(n-1)-1)))
(Magma) [n*2^(n-2)*(2^(n-1)-1): n in [0..25]];
(Maxima) makelist(n*2^(n-2)*(2^(n-1)-1), n, 0, 25);
(PARI) a(n)=n*2^(n-2)*(2^(n-1)-1);
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CROSSREFS
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Cf. A082134 (total number of edges of the pseudo-graph P).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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