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

%I A375546 #13 Sep 19 2024 08:19:52
%S A375546 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,
%T A375546 0,1,8,78,204,295,330,337,0,1,15,135,443,731,883,931,939,0,1,13,205,
%U A375546 889,1726,2275,2509,2572,2581,0,1,18,328,1813,4068,5868,6808,7148,7228,7238
%N A375546 Triangle read by rows: T(n, k) = Sum_{d|n} d * A375467(d, k) for n > 0, T(0, 0) = 1.
%e A375546 Triangle starts:
%e A375546   [0] 1;
%e A375546   [1] 0, 1;
%e A375546   [2] 0, 1,  3;
%e A375546   [3] 0, 1,  4,   7;
%e A375546   [4] 0, 1,  7,  15,  19;
%e A375546   [5] 0, 1,  6,  26,  41,   46;
%e A375546   [6] 0, 1, 12,  51,  99,  123,  129;
%e A375546   [7] 0, 1,  8,  78, 204,  295,  330,  337;
%e A375546   [8] 0, 1, 15, 135, 443,  731,  883,  931,  939;
%e A375546   [9] 0, 1, 13, 205, 889, 1726, 2275, 2509, 2572, 2581;
%p A375546 div := n -> numtheory:-divisors(n):
%p A375546 T := proc(n, k) option remember; local d; if n = 0 then 1 else
%p A375546 add(d * A375467(d, k), d = div(n)) fi end:
%p A375546 seq(seq(T(n, k), k = 0..n), n = 0..10):
%o A375546 (Python)
%o A375546 from functools import cache
%o A375546 @cache
%o A375546 def divisors(n):
%o A375546     return [d for d in range(n, 0, -1) if n % d == 0]
%o A375546 @cache
%o A375546 def T(n, k):
%o A375546     return sum(d * r(d, k) for d in divisors(n)) if n > 0 else 1
%o A375546 @cache
%o A375546 def r(n, k):
%o A375546     if n == 1: return int(k > 0)
%o A375546     return sum(r(i, k) * T(n - i, k - 1) for i in range(1, n)) // (n - 1)
%o A375546 for n in range(9): print([T(n, k) for k in range(n + 1)])
%Y A375546 Cf. A375467, A000203 (column 2), A209397 (main diagonal), A375547 (row sums).
%K A375546 nonn,tabl
%O A375546 0,6
%A A375546 _Peter Luschny_, Sep 15 2024