This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Triangle starts: [0] 1; [1] 0, 1; [2] 0, 1, 3; [3] 0, 1, 4, 7; [4] 0, 1, 7, 15, 19; [5] 0, 1, 6, 26, 41, 46; [6] 0, 1, 12, 51, 99, 123, 129; [7] 0, 1, 8, 78, 204, 295, 330, 337; [8] 0, 1, 15, 135, 443, 731, 883, 931, 939; [9] 0, 1, 13, 205, 889, 1726, 2275, 2509, 2572, 2581;
div := n -> numtheory:-divisors(n): T := proc(n, k) option remember; local d; if n = 0 then 1 else add(d * A375467(d, k), d = div(n)) fi end: seq(seq(T(n, k), k = 0..n), n = 0..10):
from functools import cache
@cache
def divisors(n):
return [d for d in range(n, 0, -1) if n % d == 0]
@cache
def T(n, k):
return sum(d * r(d, k) for d in divisors(n)) if n > 0 else 1
@cache
def r(n, k):
if n == 1: return int(k > 0)
return sum(r(i, k) * T(n - i, k - 1) for i in range(1, n)) // (n - 1)
for n in range(9): print([T(n, k) for k in range(n + 1)])
Triangle begins: 1; 1 1; 1 2 1; 1 4 3 1; 1 6 8 4 1; 1 10 18 13 5 1; 1 14 38 36 19 6 1; thus there are 10 trees with 7 nodes and height 2.
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-1, j)*b(n-i*j, i-1, k), j=0..n/i)))
end:
T:= (n, k)-> b((n-1)$2, k) -b((n-1)$2, k-1):
seq(seq(T(n, k), k=1..n-1), n=2..16); # Alois P. Heinz, Jul 31 2013
Drop[Map[Select[#, # > 0 &] &,
Transpose[
Prepend[Table[
f[n_] :=
Nest[CoefficientList[
Series[Product[1/(1 - x^i)^#[[i]], {i, 1, Length[#]}], {x,
0, 10}], x] &, {1}, n]; f[m] - f[m - 1], {m, 2, 10}],
Prepend[Table[1, {10}], 0]]]], 1] // Grid (* Geoffrey Critzer, Aug 01 2013 *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1 || k<1, 0, Sum[Binomial[b[i-1, i-1, k-1]+j-1, j]*b[n-i*j, i-1, k], {j, 0, n/i}]]]; T[n_, k_] := b[n-1, n-1, k]-b[n-1, n-1, k-1]; Table[T[n, k], {n, 2, 16}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 11 2014, after Alois P. Heinz *)
def A034781(n, k): return A375467(n, k) - A375467(n, k - 1) for n in range(2, 10): print([A034781(n, k) for k in range(2, n + 1)]) # Peter Luschny, Aug 30 2024