%I #17 Jul 24 2016 20:15:19

%S 0,0,4,31,227,1782,15564,151455,1635703,19457998,252962528,3568119351,

%T 54262590843,884831668974,15397747311556,284767367151241,

%U 5576696534340377,115269731259650802,2507575460681918232,57262481202198407625,1369461739333488200365

%N Number of graphs on n labeled nodes with maximal degree exactly 2.

%D D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.4.

%H Alois P. Heinz, <a href="/A136284/b136284.txt">Table of n, a(n) for n = 1..200</a>

%F Equals A136281 - A000085.

%F Recurrence: 2*(n-3)*(9*n-64)*a(n) = 2*(18*n^3 - 182*n^2 + 423*n - 149)*a(n-1) - 2*(n-1)*(9*n^3 - 91*n^2 + 243*n - 173)*a(n-2) + 6*(n-2)*(n-1)*(n+1)*a(n-3) + (n-3)*(n-2)*(n-1)*(9*n^2 - 91*n + 224)*a(n-4) - (n-4)*(n-3)*(n-2)*(n-1)*(9*n-67)*a(n-5) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(9*n-55)*a(n-6). - _Vaclav Kotesovec_, Feb 09 2014

%F a(n) ~ exp(sqrt(2*n)-n-1/2) * n^n / sqrt(2) * (1 + 19/(24*sqrt(2*n))). - _Vaclav Kotesovec_, Feb 09 2014

%F E.g.f.: exp(1/(1-x)/2 - 1/2 + log(1/(1-x))/2-x^2/4) - exp(x+x^2/2!). - _Joerg Arndt_, Jul 24 2016

%t nn = 20; Drop[Range[0, nn]! CoefficientList[Series[Exp[1/(1 - z)/2 - 1/2 + Log[1/(1 - z)]/2 - z^2/4] - Exp[z + z^2/2!], {z, 0, nn}], z], 1] (* _Geoffrey Critzer_, Jul 23 2016 *)

%o (PARI) x='x+O('x^22); concat( [0,0], Vec( serlaplace( exp(1/(1-x)/2 - 1/2 + log(1/(1-x))/2-x^2/4) - exp(x+x^2/2!) ) ) ) \\ _Joerg Arndt_, Jul 24 2016

%Y Cf. A000085 (degree at most 1), A136281, A136282, A136283, A136285, A136286.

%K nonn

%O 1,3

%A _Don Knuth_, Mar 31 2008

%E More terms from _Alois P. Heinz_, Sep 12 2008