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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

Table of n, a(n) for n=1..45.

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 * A319543 A319544 A319545

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, Sep 22 2018

STATUS

approved

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Last modified October 18 17:22 EDT 2018. Contains 316323 sequences. (Running on oeis4.)