A227819 - OEIS

A227819

Number T(n,k) of n-node rooted identity trees of height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 3, 1, 0, 0, 0, 2, 5, 4, 1, 0, 0, 0, 2, 8, 9, 5, 1, 0, 0, 0, 1, 12, 18, 14, 6, 1, 0, 0, 0, 1, 17, 34, 33, 20, 7, 1, 0, 0, 0, 1, 23, 61, 72, 54, 27, 8, 1, 0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1, 0, 0, 0, 0, 41, 187, 301, 303, 221, 118, 44, 10, 1

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OFFSET

1,14

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

EXAMPLE

: T(6,4) = 3 : T(11,3) = 1 :

: o o o : o :

: / \ | | : /( )\ :

: o o o o : o o o o :

: | / \ | : /| | | :

: o o o o : o o o o :

: | | / \ : | | :

: o o o o : o o :

: | | | : :

: o o o : :

Triangle T(n,k) begins:

0, 1;

0, 0, 1;

0, 0, 1, 1;

0, 0, 0, 2, 1;

0, 0, 0, 2, 3, 1;

0, 0, 0, 2, 5, 4, 1;

0, 0, 0, 2, 8, 9, 5, 1;

0, 0, 0, 1, 12, 18, 14, 6, 1;

0, 0, 0, 1, 17, 34, 33, 20, 7, 1;

0, 0, 0, 1, 23, 61, 72, 54, 27, 8, 1;

0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1;

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,

add(binomial(b((i-1)$2, k-1), j)*b(n-i*j, i-1, k), j=0..n/i)))

end:

T:= (n, k)-> b((n-1)$2, k) -`if`(k=0, 0, b((n-1)$2, k-1)):

seq(seq(T(n, k), k=0..n-1), n=1..15);

MATHEMATICA

Drop[Transpose[Map[PadRight[#, 15]&, Table[f[n_]:=Nest[ CoefficientList[ Series[ Product[(1+x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 15}], x]&, {1}, n]; f[m]-PadRight[f[m-1], Length[f[m]]], {m, 1, 15}]]], 1]//Grid (* Geoffrey Critzer, Aug 01 2013 *)

CROSSREFS

Columns k=4-10 give: A038088, A038089, A038090, A038091, A038092, A229403, A229404.

Row sums give: A004111.

Column sums give: A038081.

Largest n with T(n,k)>0 is A038093(k).

Main diagonal and lower diagonals give (offsets may differ): A000012, A001477, A000096, A166830.

T(2n,n) gives A245090.

T(2n+1,n) gives A245091.

Cf. A034781.

Sequence in context: A106278 A339829 A177517 * A064287 A196389 A128206

Adjacent sequences: A227816 A227817 A227818 * A227820 A227821 A227822

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 31 2013

STATUS

approved