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 A052327 Number of rooted trees with a forbidden limb of length 4. 3
 1, 1, 2, 4, 8, 18, 43, 102, 251, 625, 1584, 4055, 10509, 27451, 72307, 191697, 511335, 1370995, 3693452, 9991671, 27133149, 73934800, 202096673, 553999573, 1522651908, 4195087022, 11583820212, 32052475655, 88860186023 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 N. J. A. Sloane, Transforms FORMULA a(n) satisfies a = SHIFT_RIGHT(EULER(a-b)) where b(4)=1, b(k)=0 if k != 4. a(n) ~ c * d^n / n^(3/2), where d = 2.9224691962496551739365155005926289..., c = 0.43112017460637374030857983498164... . - Vaclav Kotesovec, Aug 25 2014 MAPLE with(numtheory): g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-       `if`(d=4, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)     end: a:= n-> g(n-1): seq(a(n), n=1..35);  # Alois P. Heinz, Jul 04 2014 MATHEMATICA g[n_] := g[n] = If[n == 0, 1, Sum[DivisorSum[j, #*(g[# - 1] - If[# == 4, 1, 0])&] * g[n - j], {j, 1, n}]/n]; a[n_] := g[n - 1]; Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Apr 04 2017, after Alois P. Heinz *) CROSSREFS Cf. A002955, A002988-A002992, A052318-A052329. Column k=4 of A255636. Sequence in context: A027056 A024428 A049075 * A059221 A193617 A233139 Adjacent sequences:  A052324 A052325 A052326 * A052328 A052329 A052330 KEYWORD nonn AUTHOR Christian G. Bower, Dec 15 1999 STATUS approved

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