A375546 Triangle read by rows: T(n, k) = Sum_{d|n} d * A375467(d, k) for n > 0, T(0, 0) = 1.

1, 0, 1, 0, 1, 3, 0, 1, 4, 7, 0, 1, 7, 15, 19, 0, 1, 6, 26, 41, 46, 0, 1, 12, 51, 99, 123, 129, 0, 1, 8, 78, 204, 295, 330, 337, 0, 1, 15, 135, 443, 731, 883, 931, 939, 0, 1, 13, 205, 889, 1726, 2275, 2509, 2572, 2581, 0, 1, 18, 328, 1813, 4068, 5868, 6808, 7148, 7228, 7238

Offset: 0

Author

Peter Luschny, Sep 15 2024

Keywords

• nonn
• tabl

Examples

			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;
		

Crossrefs

Cf. A375467, A000203 (column 2), A209397 (main diagonal), A375547 (row sums).

Programs

Maple

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):

Python

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)])