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 A318758 Number T(n,k) of rooted trees with n nodes such that k equals the maximal number of isomorphic subtrees extending from the same node; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows. 12
 1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 4, 1, 1, 0, 6, 9, 3, 1, 1, 0, 12, 22, 9, 3, 1, 1, 0, 25, 54, 23, 8, 3, 1, 1, 0, 52, 138, 60, 23, 8, 3, 1, 1, 0, 113, 346, 164, 61, 22, 8, 3, 1, 1, 0, 247, 889, 443, 167, 61, 22, 8, 3, 1, 1, 0, 548, 2285, 1209, 461, 168, 60, 22, 8, 3, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,8 COMMENTS T(n,k) is defined for n,k >= 0.  The triangle contains only the terms with k < n.  T(n,k) = 0 for k >= n. LINKS Alois P. Heinz, Rows n = 1..200, flattened FORMULA T(n,k) = A318757(n,k) - A318757(n,k-1) for k > 0, A(n,0) = A063524(n). EXAMPLE Triangle T(n,k) begins:   1;   0,   1;   0,   1,   1;   0,   2,   1,   1;   0,   3,   4,   1,  1;   0,   6,   9,   3,  1,  1;   0,  12,  22,   9,  3,  1, 1;   0,  25,  54,  23,  8,  3, 1, 1;   0,  52, 138,  60, 23,  8, 3, 1, 1;   0, 113, 346, 164, 61, 22, 8, 3, 1, 1; MAPLE h:= proc(n, m, t, k) option remember; `if`(m=0, binomial(n+t, t),       `if`(n=0, 0, add(h(n-1, m-j, t+1, k), j=1..min(k, m))))     end: b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,       add(b(n-i*j, i-1, k)*h(A(i, k), j, 0, k), j=0..n/i)))     end: A:= (n, k)-> `if`(n<2, n, b(n-1\$2, k)): T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)): seq(seq(T(n, k), k=0..n-1), n=1..14); MATHEMATICA h[n_, m_, t_, k_] := h[n, m, t, k] = If[m == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n - 1, m - j, t + 1, k], {j, 1, Min[k, m]}]]]; b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*h[A[i, k], j, 0, k], {j, 0, n/i}]]]; A[n_, k_] := If[n < 2, n, b[n - 1, n - 1, k]]; T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 1, 14}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, May 11 2019, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A063524, A004111 (for n>1), A318859, A318860, A318861, A318862, A318863, A318864, A318865, A318866, A318867. Row sums give A000081. T(2n+2,n+1) gives A255705. Cf. A318754, A318757. Sequence in context: A124816 A238349 A318754 * A334192 A124790 A325734 Adjacent sequences:  A318755 A318756 A318757 * A318759 A318760 A318761 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 02 2018 STATUS approved

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Last modified August 13 12:57 EDT 2022. Contains 356091 sequences. (Running on oeis4.)