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

1,3

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

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A005216 A124033 A014537 * A183911 A039765 A001091

Adjacent sequences: A136281 A136282 A136283 * A136285 A136286 A136287

Keyword

nonn

Author

Don Knuth, Mar 31 2008

Extensions

More terms from Alois P. Heinz, Sep 12 2008

Status

approved