A393142 Triangle read by rows. T(n, k) = k^(n-1)*hypergeom([1, 1 - k], [], 1/k) for k > 0; T(n, 0) = 0^n.

1, 0, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 4, 15, 34, 0, 1, 8, 45, 136, 329, 0, 1, 16, 135, 544, 1645, 4056, 0, 1, 32, 405, 2176, 8225, 24336, 60997, 0, 1, 64, 1215, 8704, 41125, 146016, 426979, 1082320, 0, 1, 128, 3645, 34816, 205625, 876096, 2988853, 8658560, 22137201

Offset

0,9

Links

Table of n, a(n) for n=0..54.

Formula

T(n, k) = (-1)^(k + 1) * k^(n - k) * Gamma(k, -k) / exp(k) for k > 0.

T(n, k) = (-1)^(n + 1) * Sum_{i=1..k}((-k)^(n - i) * (k - 1)! / (k - i)!) for k > 0.

Example

Triangle starts:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1,   1]
  [3] [0, 1,   2,    5]
  [4] [0, 1,   4,   15,    34]
  [5] [0, 1,   8,   45,   136,    329]
  [6] [0, 1,  16,  135,   544,   1645,   4056]
  [7] [0, 1,  32,  405,  2176,   8225,  24336,   60997]
  [8] [0, 1,  64, 1215,  8704,  41125, 146016,  426979, 1082320]
  [9] [0, 1, 128, 3645, 34816, 205625, 876096, 2988853, 8658560, 22137201]

Maple

T := (n, k) -> local i; ifelse(n = 0, 1, -(-1)^n * add((-k)^(n-i) * pochhammer(k-i+1, i-1), i = 1..k)): seq(print(seq(T(n, k), k = 0..n)), n = 0..9);

Mathematica

T[n_, k_] = If[k == 0, Boole[k == n], k^(n-1)*HypergeometricPFQ[{1, 1 - k}, {},  1/k]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // MatrixForm

Prog

(Python)
from math import perm
def T(n: int, k: int) -> int:
    if k == 0: return k**n
    return (-1)**(n+1) * sum((-k)**(n-i) * perm(k-1, i-1) for i in range(1, k+1))
(Python)  # efficient iterative variant
def T(n: int, k: int) -> int:
    if k == 0: return 1 if n == 0 else 0
    q, p = 1, 1
    for m in range(1, k):
        p *= -k
        q = p + m * q
    nk = 1
    for _ in range(n - k): nk *= k
    sign = 1 if k % 2 == 1 else -1
    return sign * nk * q
for n in range(10): print([T(n, k) for k in range(0, n+1)])
(Python)  # extreme edition
from mpmath import mp, gammainc, exp, nint
def T(n: int, k: int) -> int:
    if k == 0: return k**n
    mp.dps = 50
    val = (-1)**(k+1) * k**(n-k) * gammainc(k, -k) / exp(k)
    return int(nint(val))

Crossrefs

Cf. A133297, A001865, A394078 (row sums).

Sequence in context: A396973 A396499 A341268 * A011183 A005671 A343605

Adjacent sequences: A393139 A393140 A393141 * A393143 A393144 A393145

Keyword

nonn,tabl

Author

Peter Luschny, Mar 09 2026

Status

approved