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A053531
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Expansion of e.g.f.: (1-x)^(-x/2)*exp(-x^2*(4 +2*x +x^2)/8).
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2
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1, 0, 0, 0, 1, 15, 72, 420, 2915, 24570, 245070, 2633400, 30588783, 383841315, 5197243590, 75666140550, 1177491151785, 19496256883740, 342184849138188, 6346249258076280, 124023565540658025, 2547445128977720475, 54865546632888272820
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OFFSET
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0,6
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COMMENTS
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The number of simple labeled graphs on n nodes whose connected components are wheels. - Geoffrey Critzer, Dec 10 2011
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(c).
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LINKS
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FORMULA
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a(n) = n!*Sum_{m=1..n} (-2)^(-m)/m!* Sum_{k=0..m} (binomial(m,k)* Sum_{i=k..n-2*m+k} (2^(k-i)* Sum_{j=0..k} binomial(k,j)*binomial(j, i-3*k+2*j) * (-1)^(n-m-i-2*(m-k))*(m-k)!/(n-m-i)!*stirling1(n-m-i,m-k) ) ), n>0. - Vladimir Kruchinin, Sep 10 2010
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MATHEMATICA
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nn = 30; a = Sum[(n (n - 2)!/2) x^n/n!, {n, 5, nn}]; Range[0, nn]! CoefficientList[Series[Exp[x^4/4! + a], {x, 0, nn}], x] (* Geoffrey Critzer, Dec 10 2011 *)
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PROG
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(Maxima) a(n):=n!*sum((-2)^(-m)/m!*sum(binomial(m, k)*sum(2^(k-i)* sum(binomial(k, j)*binomial(j, i-3*k+2*j), j, 0, k)*(-1)^(n-m-i-2*(m-k))*(m-k)!/(n-m-i)!*stirling1(n-m-i, m-k), i, k, n-2*m+k), k, 0, m), m, 1, n); /* Vladimir Kruchinin, Sep 10 2010 */
(PARI) my(x='x+O('x^30)); Vec(serlaplace( (1-x)^(-x/2)*exp(-x^2*(4 +2*x +x^2)/8) )) \\ G. C. Greubel, May 15 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (1-x)^(-x/2)*Exp(-x^2*(4 +2*x +x^2)/8) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 15 2019
(Sage) m = 30; T = taylor((1-x)^(-x/2)*exp(-x^2*(4 +2*x +x^2)/8), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 15 2019
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CROSSREFS
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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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