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A362370 Triangle read by rows. T(n, k) = ([x^k] P(n, x)) // k! where P(n, x) = Sum_{k=1..n} P(n - k, x) * x if n >= 1 and P(0, x) = 1. The notation 's // t' means integer division and is a shortcut for 'floor(s/t)'.

%I A362370 #9 Apr 18 2023 08:29:30
%S A362370 1,0,1,0,1,0,0,1,1,0,0,1,1,0,0,0,1,2,1,0,0,0,1,2,1,0,0,0,0,1,3,2,0,0,
%T A362370 0,0,0,1,3,3,1,0,0,0,0,0,1,4,4,2,0,0,0,0,0,0,1,4,6,3,1,0,0,0,0,0,0,1,
%U A362370 5,7,5,1,0,0,0,0,0,0,0,1,5,9,6,2,0,0,0,0,0,0,0
%N A362370 Triangle read by rows. T(n, k) = ([x^k] P(n, x)) // k! where P(n, x) = Sum_{k=1..n} P(n - k, x) * x if n >= 1 and P(0, x) = 1. The notation 's // t' means integer division and is a shortcut for 'floor(s/t)'.
%C A362370 Row n gives the coefficients of the set partition polynomials of type m = 0 (the base case). The sequence of these polynomial sequences starts: this sequence, A048993, A156289, A291451, A291452, ...
%F A362370 T(n, k) = floor(A097805(n, k) / k!).
%e A362370 Triangle T(n, k) starts:
%e A362370   [0] [1]
%e A362370   [1] [0, 1]
%e A362370   [2] [0, 1, 0]
%e A362370   [3] [0, 1, 1, 0]
%e A362370   [4] [0, 1, 1, 0, 0]
%e A362370   [5] [0, 1, 2, 1, 0, 0]
%e A362370   [6] [0, 1, 2, 1, 0, 0, 0]
%e A362370   [7] [0, 1, 3, 2, 0, 0, 0, 0]
%e A362370   [8] [0, 1, 3, 3, 1, 0, 0, 0, 0]
%e A362370   [9] [0, 1, 4, 4, 2, 0, 0, 0, 0, 0]
%p A362370 T := (n, k) -> iquo(binomial(n - 1, k - 1), k!):
%p A362370 seq(print(seq(T(n, k), k = 0..n)), n = 0..9);
%o A362370 (SageMath)
%o A362370 R = PowerSeriesRing(ZZ, "x")
%o A362370 x = R.gen().O(33)
%o A362370 @cached_function
%o A362370 def p(n) -> Polynomial:
%o A362370     if n == 0: return R(1)
%o A362370     return sum(p(n - k) * x for k in range(1, n + 1))
%o A362370 def A362370_row(n) -> list[int]:
%o A362370     L = p(n).list()
%o A362370     return [L[k] // factorial(k) for k in range(n + 1)]
%o A362370 for n in range(10):
%o A362370     print(A362370_row(n))
%Y A362370 Cf. A097805, A362307 (row sums).
%Y A362370 Cf. the family of partition polynomials: this sequence (m=0), A048993 (m=1), A156289 (m=2), A291451 (m=3), A291452 (m=4).
%K A362370 nonn,tabl
%O A362370 0,18
%A A362370 _Peter Luschny_, Apr 17 2023