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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.
18
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
L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 (2018), doi SIGMA 17 (2021)
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
Rows n=0-3 give: A000004, A000012, A001477, A005449.
Lower diagonal gives A242375.
Sequence in context: A059848 A352361 A036865 * A125226 A281790 A245254
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 09 2014
STATUS
approved