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A005200
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Total number of fixed points in rooted trees with n nodes.
(Formerly M1247)
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8
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1, 2, 4, 11, 28, 78, 213, 598, 1670, 4723, 13356, 37986, 108193, 309169, 884923, 2538369, 7292170, 20982220, 60451567, 174385063, 503600439, 1455827279, 4212464112, 12199373350, 35357580112, 102552754000, 297651592188, 864460682777, 2512115979800, 7304240074858
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OFFSET
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1,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f. satisfies A(x)=T(x)[ 1+A(x)-A(x^2) ], where T(x)=x+x^2+2*x^3+... is g.f. for A000081.
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MAPLE
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# First construct T(x), the g.f. for A000081. Then we form A005200 = s and its g.f. A as follows:
s := [ 1, 2 ]; A := series(add(s[i]*x^i, i=1..2), x, 3); G := series(subs(x=x^2, A), x, 3);
for n from 3 to 30 do t1 := coeff(T, x, n)+add( coeff(T, x, i)*s[ n-i ], i=1..n-1)-add(coeff(T, x, i)*coeff(G, x, n-i), i=1..n-1); s := [ op(s), t1 ]; A := series(A+t1*x^n, x, n+1); G := series(subs(x=x^2, A), x, n+1); od: s; A;
# second Maple program:
with(numtheory): b:= proc(n) option remember; local d, j; if n<1 then 0 elif n=1 then 1 else add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1)/ (n-1) fi end: a:= proc(n) option remember; b(n) +add((b(n-i) -b(n-2*i)) *a(i), i=0..n-1) end: seq(a(n), n=1..100); # Alois P. Heinz, Sep 16 2008
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MATHEMATICA
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terms = 30; (* T = g.f. of A000081 *)
T[x_] = 0; Do[T[x_] = x*Exp[Sum[ T[x^k]/k, {k, 1, terms}]] + O[x]^(terms+1) // Normal, terms+1];
A[_] = 0; Do[A[x_] = T[x]*(1 + A[x] - A[x^2]) + O[x]^(terms+1) // Normal,
terms+1];
b[n_] := b[n] = Module[{d, j}, If[n<1, 0, If[n == 1, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}]/(n-1)]]]; a[n_] := a[n] = b[n] + Sum[ (b[n-i] - b[n-2*i])*a[i], {i, 0, n-1}]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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