A006152 Exponential generating function x*exp(x/(1-x)).

1, 2, 9, 52, 365, 3006, 28357, 301064, 3549177, 45965530, 648352001, 9888877692, 162112109029, 2841669616982, 53025262866045, 1049180850990736, 21937381717388657, 483239096122434354, 11184035897992673017, 271287473871771163460, 6881656485607798743261

Offset: 1

Author

Simon Plouffe

Keywords

• nonn
• easy

Comments

a(n) is the number of labeled rooted trees with every non-root vertex of degree 1 or 2. - Geoffrey Critzer, May 21 2012.

Total number of unit length lists in all sets of lists, cf. A000262. - Alois P. Heinz, May 10 2016

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• S. Getu and L. W. Shapiro, Combinatorial view of the composition of functions, Ars Combin. 10 (1980), 131-145. (Annotated scanned copy)
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 156

Crossrefs

Cf. A000262, A320264.

Programs

Mathematica

nn = 17; a = x/(1 - x);
Range[0, nn]! CoefficientList[Series[x Exp[a], {x, 0, nn}], x]  (* Geoffrey Critzer, May 21 2012 *)

PARI

a(n)=n!*polcoeff(x*exp(x/(1-x)+O(x^n)), n)

Formula

a(n) = n*A000262(n-1).

D-finite with recurrence a(n) = 2*(n-1)*a(n-1)-(n^2-5*n+5)*a(n-2)-(n-4)*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 05 2012

a(n) ~ n^(n-1/4)*exp^(2*sqrt(n)-n-1/2)/sqrt(2). - Vaclav Kotesovec, Oct 05 2012

a(n) = A320264(n+1,n). - Alois P. Heinz, Oct 08 2018

Extensions

More terms from Michael Somos, Jun 07 2000