A387700 Triangle read by rows: T(n, k) = binomial(2*k + n - 1, k - 2)*(n^2 - 2*k + n)/(k*(k - 1)) for k >= 2, T(n, 0) = 1 and T(n, 1) = n - 1.

1, 1, 0, 1, 1, 1, 1, 2, 4, 8, 1, 3, 8, 21, 55, 1, 4, 13, 40, 121, 364, 1, 5, 19, 66, 221, 728, 2380, 1, 6, 26, 100, 364, 1288, 4488, 15504, 1, 7, 34, 143, 560, 2108, 7752, 28101, 100947, 1, 8, 43, 196, 820, 3264, 12597, 47652, 177859, 657800

Offset

0,8

Comments

Table 3.1 in Hopkins thesis is the same below the main diagonal. - F. Chapoton, Sep 04 2025

This is a super triangle of the lower triangular matrix of A259475.

Links

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

E. Hopkins, Free Complexes over the Exterior Algebra with Small Homology, Ph.D. thesis, University of Nebraska, 2021. (Cf. Table 3.1.)

Example

Triangle begins:
   [0] [1]
   [1] [1, 0]
   [2] [1, 1,  1]
   [3] [1, 2,  4,   8]
   [4] [1, 3,  8,  21,  55]
   [5] [1, 4, 13,  40, 121,  364]
   [6] [1, 5, 19,  66, 221,  728,  2380]
   [7] [1, 6, 26, 100, 364, 1288,  4488, 15504]
   [8] [1, 7, 34, 143, 560, 2108,  7752, 28101, 100947]
   [9] [1, 8, 43, 196, 820, 3264, 12597, 47652, 177859, 657800]

Mathematica

A387700[n_, k_] := Switch[k, 0, 1, 1, n - 1, _, Binomial[2*k + n - 1, k - 2]*(n^2 - 2*k + n)/(k*(k - 1))];
Table[A387700[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Sep 07 2025 *)

Prog

(Python)
from math import comb as binomial
def T(n: int, k: int) -> int:
    if k < 0:  return 0
    if k == 0: return 1
    if k == 1: return n-1
    return binomial(2*k + n - 1, k - 2)*(n**2 - 2*k + n)//(k*(k - 1))
for n in range(9): print([T(n, k) for k in range(n+1)])

Crossrefs

Cf. A259475.

Sequence in context: A008952 A268516 A021407 * A131609 A210022 A210026

Adjacent sequences: A387697 A387698 A387699 * A387701 A387702 A387703

Keyword

nonn,tabl

Author

Peter Luschny, Sep 06 2025

Status

approved