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A227774 Triangular array read by rows: T(n,k) is the number of rooted identity trees with n nodes having exactly k subtrees from the root. 12
1, 1, 1, 1, 1, 2, 1, 3, 3, 6, 5, 1, 12, 11, 2, 25, 22, 5, 52, 49, 12, 113, 104, 28, 2, 247, 232, 65, 4, 548, 513, 152, 13, 1226, 1159, 351, 34, 2770, 2619, 818, 91, 1, 6299, 5989, 1907, 225, 6, 14426, 13734, 4460, 571, 18, 33209, 31729, 10453, 1403, 57, 76851 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,6
COMMENTS
Row sums = A004111.
LINKS
FORMULA
G.f.: x * Product_{n>=1} (1 + y * x^n)^A004111(n).
From Alois P. Heinz, Aug 25 2017: (Start)
T(n,k) = Sum_{h=0..n-k} A291529(n-1,h,k).
Sum_{k>=1} k * T(n,k) = A291532(n-1). (End)
EXAMPLE
Triangular array T(n,k) begins:
n\k: 0 1 2 3 4 ...
---+---------------------------
01 : 1;
02 : . 1;
03 : . 1;
04 : . 1, 1;
05 : . 2, 1;
06 : . 3, 3;
07 : . 6, 5, 1;
08 : . 12, 11, 2;
09 : . 25, 22, 5;
10 : . 52, 49, 12;
11 : . 113, 104, 28, 2;
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(b((i-1)$2), j)*b(n-i*j, i-1), j=0..n/i)))
end:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, expand(
add(x^j*binomial(b((i-1)$2), j)*g(n-i*j, i-1), j=0..n/i))))
end:
T:= n-> `if`(n=1, 1,
(p-> seq(coeff(p, x, k), k=1..degree(p)))(g((n-1)$2))):
seq(T(n), n=1..25); # Alois P. Heinz, Jul 30 2013
MATHEMATICA
nn=20; f[x_]:=Sum[a[n]x^n, {n, 0, nn}]; sol=SolveAlways[0==Series[f[x]-x Product[(1+x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x]; A004111=Drop[ Flatten[Table[a[n], {n, 0, nn}]/.sol], 1]; Map[Select[#, #>0&]&, Drop[CoefficientList[Series[x Product[(1 + y x^i)^A004111[[i]], {i, 1, nn}], {x, 0, nn}], {x, y}], 1]]//Grid
PROG
(Python)
from sympy import binomial, Poly, Symbol
from sympy.core.cache import cacheit
x=Symbol('x')
@cacheit
def b(n, i):return 1 if n==0 else 0 if i<1 else sum([binomial(b(i - 1, i - 1), j)*b(n - i*j, i - 1) for j in range(n//i + 1)])
@cacheit
def g(n, i):return 1 if n==0 else 0 if i<1 else sum([x**j*binomial(b(i - 1, i - 1), j)*g(n - i*j, i - 1) for j in range(n//i + 1)])
def T(n): return [1] if n==1 else Poly(g(n - 1, n - 1)).all_coeffs()[::-1][1:]
for n in range(1, 26): print(T(n)) # Indranil Ghosh, Aug 28 2017
CROSSREFS
Sequence in context: A341450 A343381 A336096 * A214920 A096373 A216961
KEYWORD
nonn,tabf,look
AUTHOR
Geoffrey Critzer, Jul 30 2013
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)