OFFSET
1,4
LINKS
FORMULA
Shifts left under "EFK" (unordered, size, unlabeled) transform.
G.f.: A(x) = x*Product_{n>=1} (1+a(n)*x^n) = Sum_{n>=1} a(n)*x^n. - Paul D. Hanna, Apr 07 2004
Lim_{n->infinity} a(n)^(1/n) = 2.5119824... - Vaclav Kotesovec, Nov 20 2019
G.f.: x * exp(Sum_{n>=1} Sum_{k>=1} (-1)^(k+1) * a(n)^k * x^(n*k) / k). - Ilya Gutkovskiy, Jun 30 2021
EXAMPLE
The a(6) = 6 fully unbalanced trees: (((((o))))), (((o(o)))), ((o((o)))), (o(((o)))), (o(o(o))), ((o)((o))). - Gus Wiseman, Jan 10 2018
MAPLE
A:= proc(n) if n<=1 then x else convert(series(x* (product(1+ coeff(A(n-1), x, i)*x^i, i=1..n-1)), x=0, n+1), polynom) fi end: a:= n-> coeff(A(n), x, n): seq(a(n), n=1..31); # Alois P. Heinz, Aug 22 2008
# second Maple program:
g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(`if`(j=0, 1, g((i-1)$2))*g(n-i*j, i-1), j=0..min(1, n/i))))
end:
a:= n-> g((n-1)$2):
seq(a(n), n=1..35); # Alois P. Heinz, Mar 04 2013
MATHEMATICA
nn=30; f[x_]:=Sum[a[n]x^n, {n, 0, nn}]; sol=SolveAlways[0 == Series[f[x]-x Product[1+a[i]x^i, {i, 1, nn}], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}]/.sol (* Geoffrey Critzer, Nov 17 2012 *)
allnim[n_]:=If[n===1, {{}}, Join@@Function[c, Select[Union[Sort/@Tuples[allnim/@c]], UnsameQ@@(Count[#, _List, {0, Infinity}]&/@#)&]]/@IntegerPartitions[n-1]];
Table[Length[allnim[n]], {n, 15}] (* Gus Wiseman, Jan 10 2018 *)
g[n_, i_] := g[n, i] = If[n == 0, 1, If[i < 1, 0,
Sum[If[j == 0, 1, g[i-1, i-1]]*g[n-i*j, i-1], {j, 0, Min[1, n/i]}]]];
a[n_] := g[n-1, n-1];
Array[a, 35] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)
PROG
(PARI) a(n)=polcoeff(x*prod(i=1, n-1, 1+a(i)*x^i)+x*O(x^n), n)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved