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 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) 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 Adjacent sequences:  A005196 A005197 A005198 * A005200 A005201 A005202 KEYWORD nonn,changed AUTHOR EXTENSIONS Definition clarified by N. J. A. Sloane, May 29 2012 STATUS approved

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Last modified August 8 02:45 EDT 2020. Contains 336290 sequences. (Running on oeis4.)