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 A255704 Number T(n,k) of n-node rooted trees in which the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root equals k; triangle T(n,k), n>=1, 1<=k<=n, read by rows. 11
 1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 3, 1, 1, 0, 17, 18, 8, 3, 1, 1, 0, 36, 45, 21, 8, 3, 1, 1, 0, 79, 116, 56, 22, 8, 3, 1, 1, 0, 175, 298, 152, 59, 22, 8, 3, 1, 1, 0, 395, 776, 413, 163, 60, 22, 8, 3, 1, 1, 0, 899, 2025, 1131, 450, 166, 60, 22, 8, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 LINKS Alois P. Heinz, Rows n = 1..141, flattened FORMULA T(n,1) = A255636(n,1), T(n,k) = A255636(n,k) - A255636(n,k-1) for k>1. EXAMPLE :    o      o     o         o     o     o     o :  /( )\   /|\   / \       / \    |     |     | : o o o o o o o o   o     o   o   o     o     o : |       | |   |  / \   / \     /|\   / \    | : o       o o   o o   o o   o   o o o o   o   o :                       |       |     |   |  / \ :                       o       o     o   o o   o :                                           | : T(6,3) = 7                                o Triangle T(n,k) begins:   1;   0,   1;   0,   1,   1;   0,   2,   1,   1;   0,   4,   3,   1,  1;   0,   8,   7,   3,  1,  1;   0,  17,  18,   8,  3,  1, 1;   0,  36,  45,  21,  8,  3, 1, 1;   0,  79, 116,  56, 22,  8, 3, 1, 1;   0, 175, 298, 152, 59, 22, 8, 3, 1, 1; MAPLE with(numtheory): g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-       `if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)     end: T:= (n, k)-> g(n-1, k) -`if`(k=1, 0, g(n-1, k-1)): seq(seq(T(n, k), k=1..n), n=1..14); MATHEMATICA g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*(g[#-1, k] - If[# == k, 1, 0])&] * g[n-j, k], {j, 1, n}]/n]; T[n_, k_] := g[n-1, k] - If[k == 1, 0, g[n-1, k-1]]; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 24 2017, translated from Maple *) CROSSREFS Columns k=1-10 give: A063524, A002955 (for n>1), A318899, A318900, A318901, A318902, A318903, A318904, A318905, A318906. Row sums give A000081. T(2*n+1,n+1) gives A255705. Cf. A255636. Sequence in context: A135221 A318686 A214546 * A191347 A106234 A238125 Adjacent sequences:  A255701 A255702 A255703 * A255705 A255706 A255707 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Mar 02 2015 STATUS approved

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Last modified November 28 19:19 EST 2021. Contains 349415 sequences. (Running on oeis4.)