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A053525
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Expansion of e.g.f.: (1-x)/(2-exp(x)).
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7
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1, 0, 1, 4, 23, 166, 1437, 14512, 167491, 2174746, 31374953, 497909380, 8619976719, 161667969646, 3265326093109, 70663046421208, 1631123626335707, 40004637435452866, 1038860856732399105, 28476428717448349996, 821656049857815980455
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OFFSET
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0,4
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COMMENTS
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The number of connected labeled threshold graphs on n vertices. - Sam Spiro, Sep 22 2019
Also the number of 2-interval parking functions of size n. - Sam Spiro, Sep 24 2019
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.4(a).
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LINKS
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Jean-Christophe Aval, Adrien Boussicault, Philippe Nadeau, Tree-like Tableaux, Electronic Journal of Combinatorics, 20(4), 2013, #P34.
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FORMULA
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a(n) = c(n) - n*c(n-1) where c() = A000670.
a(n) = Sum_{k=0..n-1} binomial(n, k) * a(k), n>1. - Michael Somos, Aug 01 2016
E.g.f. A(x) satisfies (1 - x) * A'(x) = A(x) * (x - 2 + 2*A(x)). - Michael Somos, Aug 01 2016
a(n) = Sum_{k=1..n-1} (n-k)*A008292(n-1,k-1)*2^(k-1), valid for n>=2. - Sam Spiro, Sep 22 2019
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EXAMPLE
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G.f. = 1 + x^2 + 4*x^3 + 23*x^4 + 166*x^5 + 1437*x^6 + 14512*x^7 + ...
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MAPLE
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A053525 := proc(n) option remember;
`if`(n < 2, 1 - n, add(binomial(n, k) * A053525(k), k = 0..n-1)) end:
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MATHEMATICA
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With[{nn=20}, CoefficientList[Series[(1-x)/(2-Exp[x]), {x, 0, nn}], x] Range[0, nn]!] Harvey P. Dale, May 17 2012
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PROG
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(PARI) {a(n) = if( n<0, 0, n! * polcoeff( (1 - x) / (2 - exp(x + x*O(x^n))), n))}; /* Michael Somos, Aug 01 2016 */
(Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)/(2-Exp(x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Mar 15 2019
(Sage) m = 25; T = taylor((1-x)/(2-exp(x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Mar 15 2019
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CROSSREFS
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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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