A136284 Number of graphs on n labeled nodes with maximal degree exactly 2.

0, 0, 4, 31, 227, 1782, 15564, 151455, 1635703, 19457998, 252962528, 3568119351, 54262590843, 884831668974, 15397747311556, 284767367151241, 5576696534340377, 115269731259650802, 2507575460681918232, 57262481202198407625, 1369461739333488200365

Offset: 1

Author

Don Knuth, Mar 31 2008

Keywords

• nonn

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200

Crossrefs

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

Programs

Mathematica

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 *)

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

Formula

Equals A136281 - A000085.

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

a(n) ~ exp(sqrt(2*n)-n-1/2) * n^n / sqrt(2) * (1 + 19/(24*sqrt(2*n))). - Vaclav Kotesovec, Feb 09 2014

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

Extensions

More terms from Alois P. Heinz, Sep 12 2008