login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005198 a(n) is the number of forests with n (unlabeled) nodes in which each component tree is planted, that is, is a rooted tree in which the root has degree 1.
(Formerly M2491)
1
0, 1, 1, 3, 5, 13, 27, 68, 160, 404, 1010, 2604, 6726, 17661, 46628, 124287, 333162, 898921, 2437254, 6640537, 18166568, 49890419, 137478389, 380031868, 1053517588, 2928246650, 8158727139, 22782938271, 63752461474, 178740014515, 502026565792, 1412409894224 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

REFERENCES

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..2136 (first 118 terms from Washington Bomfim)

E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121.

FORMULA

a(1) = 0, if n >= 2 a(n) = Sum_{P_1(n)}( Product_{k=2..n} binomial(A000081(k-1) + c_k - 1, c_k) ), where P_1(n) are the partitions of n without parts equal to 1: 2*c_2 + ... + n*c_n = n; c_2, ..., c_n >= 0. - Washington Bomfim, Jul 05 2020

MAPLE

g:= proc(n) option remember; `if`(n<=1, n, (add(add(d*g(d),

       d=numtheory[divisors](j))*g(n-j), j=1..n-1))/(n-1))

    end:

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<2, 0, add(

       binomial(g(i-1)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))

    end:

a:= n-> b(n$2):

seq(a(n), n=1..40);  # Alois P. Heinz, Jul 07 2020

PROG

(PARI) g(m) = {my(f); if(m==0, return(1)); f = vector(m+1); f[1]=1;

for(j=1, m, f[j+1]=1/j * sum(k=1, j, sumdiv(k, d, d * f[d]) * f[j-k+1])); f[m+1] };

global(max_n = 130); A000081 = vector(max_n, n, g(n-1));

seq(n)={my(s=0, D, c, P_1); if(n==1, return(0)); forpart(P_1 = n, D = Set(P_1); c = vector(#D); for(k=1, #D, c[k] = #select(x->x == D[k], Vec(P_1)));

s += prod(k=1, #D, binomial( A000081[D[k]-1] + c[k] - 1, c[k]) ), [2, n], [1, n]); s}; \\ Washington Bomfim, Jul 05 2020

CROSSREFS

Cf. A000081.

Sequence in context: A000631 A026569 A035082 * A160823 A077443 A147196

Adjacent sequences:  A005195 A005196 A005197 * A005199 A005200 A005201

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Definition clarified and more terms added from Palmer-Schwenk by N. J. A. Sloane, May 29 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 15 01:47 EDT 2020. Contains 336485 sequences. (Running on oeis4.)