A005199 - OEIS

A005199

a(n) = Sum_t t*F(n,t), where F(n,t) is the number of forests with n (unlabeled) nodes and exactly t trees, all of which are planted (that is, rooted trees in which the root has degree 1).
(Formerly M3285)

0, 1, 1, 4, 6, 18, 35, 93, 214, 549, 1362, 3534, 9102, 23951, 63192, 168561, 451764, 1219290, 3305783, 9008027, 24643538, 67681372, 186504925, 515566016, 1429246490, 3972598378, 11068477743, 30908170493, 86488245455, 242481159915, 681048784377, 1916051725977, 5399062619966

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OFFSET

1,4

COMMENTS

The triangular array F(n,t) (analogous to A095133 for A005196 and A033185 for A005197) is A336087.

REFERENCES

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

LINKS

Washington Bomfim, Table of n, a(n) for n = 1..120

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(n) = Sum_{t=1, floor(n/2)}( t*F(n,t) ), where F(n,t) = Sum_{P_1(n,t)} (Product_{k=2..n} binomial(A000081(k-1) + c_k - 1, c_k)), where P_1(n, t) is the set of the partitions of n with t parts greater than one: 2*c_2 + ... + n*c_n = n; c_2, ..., c_n >= 0. - Washington Bomfim, Jul 08 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));

F(n, t)={my(s=0, D, c, P_1); 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], [t, t]); s};

seq(n) = sum(t=1, n\2, t*F(n, t) ); \\ Washington Bomfim, Jul 08 2020

CROSSREFS

Cf. A000081, A336087.

Sequence in context: A281861 A218898 A088810 * A107390 A051253 A175955