A395701 Triangle read by rows: T(n, k) = binomial(n, k) * k^k.

1, 1, 1, 1, 2, 4, 1, 3, 12, 27, 1, 4, 24, 108, 256, 1, 5, 40, 270, 1280, 3125, 1, 6, 60, 540, 3840, 18750, 46656, 1, 7, 84, 945, 8960, 65625, 326592, 823543, 1, 8, 112, 1512, 17920, 175000, 1306368, 6588344, 16777216, 1, 9, 144, 2268, 32256, 393750, 3919104, 29647548, 150994944, 387420489

Offset

0,5

Links

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).

Example

Triangle starts:
  [0] [1]
  [1] [1, 1]
  [2] [1, 2,  4]
  [3] [1, 3,  12,   27]
  [4] [1, 4,  24,  108,   256]
  [5] [1, 5,  40,  270,  1280,   3125]
  [6] [1, 6,  60,  540,  3840,  18750,   46656]
  [7] [1, 7,  84,  945,  8960,  65625,  326592,   823543]
  [8] [1, 8, 112, 1512, 17920, 175000, 1306368,  6588344,  16777216]
  [9] [1, 9, 144, 2268, 32256, 393750, 3919104, 29647548, 150994944, 387420489]

Maple

T := (n, k) -> binomial(n, k) * k^k: seq(print(seq(T(n, k), k = 0..n)), n = 0..9);

Mathematica

A395701[n_, k_] := If[k == 0, 1, Binomial[n, k]*k^k];
Table[A395701[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, May 10 2026 *)

Prog

(Python)
def A395701triangle(size: int) -> list[list[int]]:
    tri = []
    mm = [1] + [m ** m for m in range(1, size+1)]
    for n in range(size):
        Qrow = [0]*(n+1)
        binom = 1
        for m in range(n+1):
            if m > 0: binom = binom * (n - m + 1) // m
            Qrow[m] = binom * mm[m]
        tri.append(Qrow)
    return tri
for row in A395701triangle(9): print(row)

Crossrefs

Cf. A000312, A007318, A086331 (row sums), A367271 (central terms).

Sequence in context: A117297 A112973 A162303 * A128570 A371767 A157284

Adjacent sequences: A395698 A395699 A395700 * A395702 A395703 A395704

Keyword

nonn,tabl

Author

Peter Luschny, May 08 2026

Status

approved