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 A319541 Triangle read by rows: T(n,k) is the number of binary rooted trees with n leaves of exactly k colors and all non-leaf nodes having out-degree 2. 10
 1, 1, 1, 1, 4, 3, 2, 14, 27, 15, 3, 48, 180, 240, 105, 6, 171, 1089, 2604, 2625, 945, 11, 614, 6333, 24180, 42075, 34020, 10395, 23, 2270, 36309, 207732, 554820, 755370, 509355, 135135, 46, 8518, 207255, 1710108, 6578550, 13408740, 14963130, 8648640, 2027025 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS See table 2.2 in the Johnson reference. LINKS Alois P. Heinz, Rows n = 1..141, flattened V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012. FORMULA T(n,k) = Sum_{i=1..k} (-1)^(k-i)*binomial(k,i)*A319539(n,i). EXAMPLE Triangle begins:    1;    1,    1;    1,    4,     3;    2,   14,    27,     15;    3,   48,   180,    240,    105;    6,  171,  1089,   2604,   2625,    945;   11,  614,  6333,  24180,  42075,  34020,  10395;   23, 2270, 36309, 207732, 554820, 755370, 509355, 135135;   ... MAPLE A:= proc(n, k) option remember; `if`(n<2, k*n, `if`(n::odd, 0,       (t-> t*(1-t)/2)(A(n/2, k)))+add(A(i, k)*A(n-i, k), i=1..n/2))     end: T:= (n, k)-> add((-1)^i*binomial(k, i)*A(n, k-i), i=0..k): seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Sep 23 2018 MATHEMATICA A[n_, k_] := A[n, k] = If[n<2, k n, If[OddQ[n], 0, (#(1-#)/2)&[A[n/2, k]]] + Sum[A[i, k] A[n - i, k], {i, 1, n/2}]]; T[n_, k_] := Sum[(-1)^i Binomial[k, i] A[n, k - i], {i, 0, k}]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 02 2019, after Alois P. Heinz *) PROG (PARI) \\ here R(n, k) is k-th column of A319539 as a vector. R(n, k)={my(v=vector(n)); v[1]=k; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v} M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))} {my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))} CROSSREFS Columns 1..5 are A001190, A220819, A220820, A220821, A220822. Main diagonal is A001147. Row sums give A319590. Cf. A241555, A319376, A319539. Sequence in context: A099406 A274601 A202696 * A239020 A293211 A330778 Adjacent sequences:  A319538 A319539 A319540 * A319542 A319543 A319544 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, Sep 22 2018 STATUS approved

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Last modified September 21 13:00 EDT 2020. Contains 337272 sequences. (Running on oeis4.)