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)'.

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

Offset: 0

Author

Peter Luschny, Apr 17 2023

Keywords

• nonn
• tabl

Comments

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, ...

Examples

			Triangle T(n, k) starts:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1, 0]
  [3] [0, 1, 1, 0]
  [4] [0, 1, 1, 0, 0]
  [5] [0, 1, 2, 1, 0, 0]
  [6] [0, 1, 2, 1, 0, 0, 0]
  [7] [0, 1, 3, 2, 0, 0, 0, 0]
  [8] [0, 1, 3, 3, 1, 0, 0, 0, 0]
  [9] [0, 1, 4, 4, 2, 0, 0, 0, 0, 0]
		

Crossrefs

Cf. A097805, A362307 (row sums).

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

Programs

Maple

T := (n, k) -> iquo(binomial(n - 1, k - 1), k!):
seq(print(seq(T(n, k), k = 0..n)), n = 0..9);

SageMath

R = PowerSeriesRing(ZZ, "x")
x = R.gen().O(33)
@cached_function
def p(n) -> Polynomial:
    if n == 0: return R(1)
    return sum(p(n - k) * x for k in range(1, n + 1))
def A362370_row(n) -> list[int]:
    L = p(n).list()
    return [L[k] // factorial(k) for k in range(n + 1)]
for n in range(10):
    print(A362370_row(n))

Formula

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