A242249 Number A(n,k) of rooted trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 0, 0, 1, 3, 7, 4, 0, 0, 1, 4, 15, 26, 9, 0, 0, 1, 5, 26, 82, 107, 20, 0, 0, 1, 6, 40, 188, 495, 458, 48, 0, 0, 1, 7, 57, 360, 1499, 3144, 2058, 115, 0, 0, 1, 8, 77, 614, 3570, 12628, 20875, 9498, 286, 0, 0, 1, 9, 100, 966, 7284, 37476, 111064, 142773, 44947, 719, 0

Offset

0,13

Comments

From Vaclav Kotesovec, Aug 26 2014: (Start)

Column k > 0 is asymptotic to c(k) * d(k)^n / n^(3/2), where constants c(k) and d(k) are dependent only on k. Conjecture: d(k) ~ k * exp(1). Numerically:

d(1) = 2.9557652856519949747148175... (A051491)

d(2) = 5.6465426162329497128927135... (A245870)

d(3) = 8.3560268792959953682760695...

d(4) = 11.0699628777593263124193026...

d(5) = 13.7856511100846851989303249...

d(6) = 16.5022088446930015657112211...

d(7) = 19.2192613290638657575973462...

d(8) = 21.9366222112987115910888213...

d(9) = 24.6541883249893084812976812...

d(10) = 27.3718979186642404090999595...

d(100) = 272.0126359583480733207362718...

d(101) = 274.7309127032967881125015217...

d(200) = 543.8405620978790523837823296...

d(201) = 546.5588426492458787468860222...

d(101)-d(100) = 2.718276744...

d(201)-d(200) = 2.718280551...

(End)

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 (2018)

Formula

G.f. for column k: x*Product_{n>=1} 1/(1 - x^n)^(k*A(n,k)). - Geoffrey Critzer, Nov 13 2014

Example

Square array A(n,k) begins:
  0,  0,    0,     0,      0,      0,       0,       0, ...
  1,  1,    1,     1,      1,      1,       1,       1, ...
  0,  1,    2,     3,      4,      5,       6,       7, ...
  0,  2,    7,    15,     26,     40,      57,      77, ...
  0,  4,   26,    82,    188,    360,     614,     966, ...
  0,  9,  107,   495,   1499,   3570,    7284,   13342, ...
  0, 20,  458,  3144,  12628,  37476,   91566,  195384, ...
  0, 48, 2058, 20875, 111064, 410490, 1200705, 2984142, ...

Maple

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*
      A(d, k), d=divisors(j))*A(n-j, k)*k, j=1..n-1))/(n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

nn = 10; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; Transpose[ Table[Flatten[ sol = SolveAlways[ 0 == Series[ t[x] - x Product[1/(1 - x^i)^(n a[i]), {i, 1, nn}], {x, 0, nn}], x]; Flatten[{0, Table[a[n], {n, 1, nn}]}] /. sol], {n, 0, nn}]] // Grid (* Geoffrey Critzer, Nov 11 2014 *)
A[n_, k_] := A[n, k] = If[n<2, n, Sum[Sum[d*A[d, k], {d, Divisors[j]}] *A[n-j, k]*k, {j, 1, n-1}]/(n-1)]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 04 2014, translated from Maple *)

Prog

(PARI) \\ ColGf gives column generating function
ColGf(N, k) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = k/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); x*Ser(A)}
Mat(vector(8, k, concat(0, Col(ColGf(7, k-1))))) \\ Andrew Howroyd, May 12 2018

Crossrefs

Columns k=0-10 give: A063524, A000081, A000151, A006964, A052763, A052788, A246235, A246236, A246237, A246238, A246239.

Rows n=0-3 give: A000004, A000012, A001477, A005449.

Lower diagonal gives A242375.

Cf. A255517, A256064.

Sequence in context: A059848 A352361 A036865 * A125226 A281790 A245254

Adjacent sequences: A242246 A242247 A242248 * A242250 A242251 A242252

Keyword

nonn,tabl

Author

Alois P. Heinz, May 09 2014

Status

approved