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

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