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 A244454 Number T(n,k) of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows. 13
 1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 7, 1, 0, 1, 0, 17, 2, 0, 0, 1, 0, 42, 4, 1, 0, 0, 1, 0, 105, 7, 2, 0, 0, 0, 1, 0, 267, 15, 2, 1, 0, 0, 0, 1, 0, 684, 28, 4, 2, 0, 0, 0, 0, 1, 0, 1775, 56, 7, 2, 1, 0, 0, 0, 0, 1, 0, 4639, 110, 12, 2, 2, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS T(1,0) = 1 by convention. Sum_{i=2..n-1} T(n,i) = A001678(n+1) for n>1. LINKS Alois P. Heinz, Rows n = 1..141, flattened EXAMPLE The A000081(5) = 9 rooted trees with 5 nodes sorted by minimal outdegree of inner nodes are: : 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   o o     o                       :         :         : : |                                     :         :         : : o                                     :         :         : :                                       :         :         : : ------------------1------------------ : ---2--- : ---4--- : Thus row 5 = [0, 7, 1, 0, 1]. Triangle T(n,k) begins:   1;   0,    1;   0,    1,   1;   0,    3,   0,  1;   0,    7,   1,  0, 1;   0,   17,   2,  0, 0, 1;   0,   42,   4,  1, 0, 0, 1;   0,  105,   7,  2, 0, 0, 0, 1;   0,  267,  15,  2, 1, 0, 0, 0, 1;   0,  684,  28,  4, 2, 0, 0, 0, 0, 1;   0, 1775,  56,  7, 2, 1, 0, 0, 0, 0, 1;   0, 4639, 110, 12, 2, 2, 0, 0, 0, 0, 0, 1; MAPLE b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],       1, 0), `if`(i<1, 0, add(binomial(b((i-1)\$2, k\$2)+j-1, j)*       b(n-i*j, i-1, max(0, t-j), k), j=0..n/i)))     end: T:= (n, k)-> b(n-1\$2, k\$2) -`if`(n=1 and k=0, 0, b(n-1\$2, k+1\$2)): seq(seq(T(n, k), k=0..n-1), n=1..14); MATHEMATICA b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i<1, 0, Sum[Binomial[b[i-1, i-1, k, k]+j-1, j]* b[n-i*j, i-1, Max[0, t-j], k], {j, 0, n/i}]]]; T[n_, k_] := b[n-1, n-1, k, k] - If[n == 1 && k == 0, 0, b[n-1, n-1, k+1, k+1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 08 2015, translated from Maple *) CROSSREFS Columns k=0-10 give: A063524, A244455, A244456, A244457, A244458, A244459, A244460, A244461, A244462, A244463, A244464. Row sums give A000081. Cf. A001678, A244372, A244530 (ordered unlabeled rooted trees). Sequence in context: A262964 A135481 A180049 * A238123 A128311 A334076 Adjacent sequences:  A244451 A244452 A244453 * A244455 A244456 A244457 KEYWORD nonn,tabl AUTHOR Joerg Arndt and Alois P. Heinz, Jun 28 2014 STATUS approved

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Last modified August 11 23:45 EDT 2020. Contains 336434 sequences. (Running on oeis4.)