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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. 9
 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 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 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. Cf. A241555, A319376, A319539. Sequence in context: A099406 A274601 A202696 * A239020 A293211 A061312 Adjacent sequences:  A319537 A319538 A319539 * A319542 A319543 A319544 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, Sep 22 2018 STATUS approved

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Last modified December 15 21:16 EST 2018. Contains 318154 sequences. (Running on oeis4.)