A362370 - OEIS

%I #9 Apr 18 2023 08:29:30

%S 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 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 5,7,5,1,0,0,0,0,0,0,0,1,5,9,6,2,0,0,0,0,0,0,0

%N 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 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 T(n, k) = floor(A097805(n, k) / k!).

%e Triangle T(n, k) starts:

%e   [0] [1]

%e   [1] [0, 1]

%e   [2] [0, 1, 0]

%e   [3] [0, 1, 1, 0]

%e   [4] [0, 1, 1, 0, 0]

%e   [5] [0, 1, 2, 1, 0, 0]

%e   [6] [0, 1, 2, 1, 0, 0, 0]

%e   [7] [0, 1, 3, 2, 0, 0, 0, 0]

%e   [8] [0, 1, 3, 3, 1, 0, 0, 0, 0]

%e   [9] [0, 1, 4, 4, 2, 0, 0, 0, 0, 0]

%p T := (n, k) -> iquo(binomial(n - 1, k - 1), k!):

%p seq(print(seq(T(n, k), k = 0..n)), n = 0..9);

%o (SageMath)

%o R = PowerSeriesRing(ZZ, "x")

%o x = R.gen().O(33)

%o @cached_function

%o def p(n) -> Polynomial:

%o     if n == 0: return R(1)

%o     return sum(p(n - k) * x for k in range(1, n + 1))

%o def A362370_row(n) -> list[int]:

%o     L = p(n).list()

%o     return [L[k] // factorial(k) for k in range(n + 1)]

%o for n in range(10):

%o     print(A362370_row(n))

%Y Cf. A097805, A362307 (row sums).

%Y Cf. the family of partition polynomials: this sequence (m=0), A048993 (m=1), A156289 (m=2), A291451 (m=3), A291452 (m=4).

%K nonn,tabl

%O 0,18

%A _Peter Luschny_, Apr 17 2023