A006152 Exponential generating function x*exp(x/(1-x)). (Formerly M1939)

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

Offset

1,2

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

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000262, A320264.

Sequence in context: A305987 A355789 A231494 * A369551 A143508 A052882

Adjacent sequences: A006149 A006150 A006151 * A006153 A006154 A006155

Keyword

nonn,easy

Author

Simon Plouffe

Extensions

More terms from Michael Somos, Jun 07 2000

Status

approved