A378322 Triangle read by rows: T(n,k) is the number of permutations (x_1, x_2, ..., x_k) such that 1/x_1 + 2/x_2 + ... + k/x_k = k, where x_1, x_2, ... x_k are distinct values between 1 and n.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 4, 1, 1, 1, 3, 3, 4, 1, 1, 1, 1, 3, 5, 10, 11, 12, 1, 1, 1, 3, 6, 16, 40, 54, 30, 1, 1, 1, 3, 7, 25, 88, 184, 206, 102, 1, 1, 1, 3, 7, 25, 88, 184, 206, 102, 1, 1, 1, 1, 3, 13, 56, 244, 744, 1329, 1623, 746, 749, 1, 1, 1, 3, 13, 56, 244, 744, 1329, 1623, 746, 749, 1, 1

Offset

1,9

Links

Table of n, a(n) for n=1..91.

Example

Triangle T(n,k) begins:
n\k | 1  2  3  4   5  6   7   8  9
----+------------------------------
  1 | 1;
  2 | 1, 1;
  3 | 1, 1, 1;
  4 | 1, 1, 2, 1;
  5 | 1, 1, 2, 1,  1;
  6 | 1, 1, 3, 3,  4,  1;
  7 | 1, 1, 3, 3,  4,  1,  1;
  8 | 1, 1, 3, 5, 10, 11, 12,  1;
  9 | 1, 1, 3, 6, 16, 40, 54, 30, 1;
  ...
T(9,1)=1  1/1 = 1
T(9,2)=1  1/1 + 2/2 = 2
T(9,3)=3  1/1 + 2/2 + 3/3 = 3
          1/1 + 2/4 + 3/2 = 3
          1/2 + 2/1 + 3/6 = 3
T(9,4)=6  1/1 + 2/2 + 3/3 + 4/4 = 4
          1/1 + 2/3 + 3/9 + 4/2 = 4
          1/1 + 2/4 + 3/6 + 4/2 = 4
          1/1 + 2/8 + 3/4 + 4/2 = 4
          1/2 + 2/1 + 3/3 + 4/8 = 4
          1/2 + 2/1 + 3/6 + 4/4 = 4

Prog

(Julia) using Combinatorics
function T(n, k)
    X = [i for i =1:n] |> x->permutations(x, k)
    Y = []
    for x ∈ X
        if sum(i//x[i] for i=1:k) == k
            push!(Y, x)
        end
    end
    Y |> length
end
(PARI) T(n, k)={my(c=0); forsubset([n, k], v, forperm(v, p, if(sum(i=1, #p, i/p[i])==k, c++))); c} \\ Andrew Howroyd, Nov 23 2024
(Python)
from math import isqrt, comb, prod
from itertools import combinations, permutations
def A378322(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    b = n-comb(a, 2)
    c = 0
    for p in combinations(range(1, a+1), b):
        q = prod(p)
        t = [(i+1)*q for i in range(b)]
        for r in permutations(p, b):
            if sum(t[i]//r[i] for i in range(b))==t[b-1]:
                c += 1
    return c # Chai Wah Wu, Dec 03 2024

Crossrefs

Sequence in context: A146289 A214575 A081418 * A088951 A361633 A186006

Adjacent sequences: A378319 A378320 A378321 * A378323 A378324 A378325

Keyword

nonn,tabl

Author

Dan Shimizu, Nov 23 2024

Extensions

More terms (rows n=10-13) from Alois P. Heinz, Nov 24 2024

Status

approved