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 A005197 a(n) = Sum_t t*F(n,t), where F(n,t) (see A033185) is the number of rooted forests with n (unlabeled) nodes and exactly t rooted trees. (Formerly M2663) 5
 1, 3, 7, 17, 39, 96, 232, 583, 1474, 3797, 9864, 25947, 68738, 183612, 493471, 1334143, 3624800, 9893860, 27113492, 74577187, 205806860, 569678759, 1581243203, 4400193551, 12273287277, 34307646762, 96093291818, 269654004899, 758014312091, 2134300171031 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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..600 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 To get a(n), take row n of the triangle in A033185, multiply successive terms by 1, 2, 3, ... and sum. E.g. a(4) = 1*4+2*3+3*1+4*1 = 17. a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.955765285..., c = 2.85007275... . - Vaclav Kotesovec, Sep 10 2014 MAPLE with(numtheory): t:= proc(n) option remember; local d, j; `if`(n<=1, n,       (add(add(d*t(d), d=divisors(j))*t(n-j), j=1..n-1))/(n-1))     end: b:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,       `if`(min(i, p)<1, 0, add(b(n-i*j, i-1, p-j) *        binomial(t(i)+j-1, j), j=0..min(n/i, p)))))     end: a:= a-> add(k*b(n, n, k), k=1..n): seq(a(n), n=1..40);  # Alois P. Heinz, Aug 20 2012 MATHEMATICA t[1] = 1; t[n_] := t[n] = Module[{d, j}, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n-j], {j, 1, n-1}]/(n-1)]; b[1, 1, 1] = 1; b[n_, i_, p_] := b[n, i, p] = If[p>n, 0, If[n == 0, 1, If[Min[i, p]<1, 0, Sum[b[n-i*j, i-1, p-j]*Binomial[t[i]+j-1, j], {j, 0, Min[n/i, p]}]]]]; a[n_] := Sum[k*b[n, n, k], {k, 1, n}]; Table[a[n] // FullSimplify, {n, 1, 30}] (* Jean-François Alcover, Mar 13 2014, after Alois P. Heinz *) CROSSREFS Cf. A000081, A005196, A033185. Sequence in context: A191825 A229514 A077927 * A147142 A298371 A106472 Adjacent sequences:  A005194 A005195 A005196 * A005198 A005199 A005200 KEYWORD nonn AUTHOR N. J. A. Sloane. Definition clarified by N. J. A. Sloane, May 29 2012 EXTENSIONS More terms from Alois P. Heinz, Aug 20 2012 STATUS approved

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Last modified January 22 15:57 EST 2019. Contains 319364 sequences. (Running on oeis4.)