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%I #28 Sep 08 2022 08:45:00
%S 1,0,0,0,1,15,72,420,2915,24570,245070,2633400,30588783,383841315,
%T 5197243590,75666140550,1177491151785,19496256883740,342184849138188,
%U 6346249258076280,124023565540658025,2547445128977720475,54865546632888272820
%N Expansion of e.g.f.: (1-x)^(-x/2)*exp(-x^2*(4 +2*x +x^2)/8).
%C The number of simple labeled graphs on n nodes whose connected components are wheels. - _Geoffrey Critzer_, Dec 10 2011
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.15(c).
%H G. C. Greubel, <a href="/A053531/b053531.txt">Table of n, a(n) for n = 0..445</a>
%H Vladimir Kruchinin, D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2013.
%F 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
%t 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 *)
%o (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 */
%o (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
%o (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
%o (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
%K nonn
%O 0,6
%A _N. J. A. Sloane_, Jan 16 2000
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