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 A299038 Number A(n,k) of rooted trees with n nodes where each node has at most k children; square array A(n,k), n>=0, k>=0, read by antidiagonals. 12
 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 3, 1, 0, 1, 1, 1, 2, 4, 6, 1, 0, 1, 1, 1, 2, 4, 8, 11, 1, 0, 1, 1, 1, 2, 4, 9, 17, 23, 1, 0, 1, 1, 1, 2, 4, 9, 19, 39, 46, 1, 0, 1, 1, 1, 2, 4, 9, 20, 45, 89, 98, 1, 0, 1, 1, 1, 2, 4, 9, 20, 47, 106, 211, 207, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,19 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened FORMULA A(n,k) = Sum_{i=0..k} A244372(n,i) for n>0, A(0,k) = 1. EXAMPLE Square array A(n,k) begins:   1, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...   1, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...   0, 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...   0, 1,   2,   2,   2,   2,   2,   2,   2,   2,   2, ...   0, 1,   3,   4,   4,   4,   4,   4,   4,   4,   4, ...   0, 1,   6,   8,   9,   9,   9,   9,   9,   9,   9, ...   0, 1,  11,  17,  19,  20,  20,  20,  20,  20,  20, ...   0, 1,  23,  39,  45,  47,  48,  48,  48,  48,  48, ...   0, 1,  46,  89, 106, 112, 114, 115, 115, 115, 115, ...   0, 1,  98, 211, 260, 277, 283, 285, 286, 286, 286, ...   0, 1, 207, 507, 643, 693, 710, 716, 718, 719, 719, ... MAPLE b:= proc(n, i, t, k) option remember; `if`(n=0, 1,       `if`(i<1, 0, add(binomial(b((i-1)\$2, k\$2)+j-1, j)*        b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))     end: A:= (n, k)-> `if`(n=0, 1, b(n-1\$2, k\$2)): seq(seq(A(n, d-n), n=0..d), d=0..14); MATHEMATICA b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[ b[i-1, i-1, k, k]+j-1, j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]]; A[n_, k_] := If[n == 0, 1, b[n - 1, n - 1, k, k]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 04 2018, from Maple *) PROG (Python) from sympy import binomial from sympy.core.cache import cacheit @cacheit def b(n, i, t, k): return 1 if n==0 else 0 if i<1 else sum([binomial(b(i-1, i-1, k, k)+j-1, j)*b(n-i*j, i-1, t-j, k) for j in range(min(t, n//i)+1)]) def A(n, k): return 1 if n==0 else b(n-1, n-1, k, k) for d in range(15): print([A(n, d-n) for n in range(d+1)]) # Indranil Ghosh, Mar 02 2018, after Maple code CROSSREFS Columns k=1-11 give: A000012, A001190(n+1), A000598, A036718, A036721, A036722, A182378, A292553, A292554, A292555, A292556. Main diagonal gives A000081 for n>0. A(2n,n) gives A299039. Cf. A244372. Sequence in context: A054635 A003137 A006842 * A273693 A219967 A060505 Adjacent sequences:  A299035 A299036 A299037 * A299039 A299040 A299041 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Feb 01 2018 STATUS approved

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Last modified September 22 11:13 EDT 2018. Contains 315270 sequences. (Running on oeis4.)