A386363 Variation of triangle of Entringer numbers (A008281) read by rows: T(n, k) = T(n, k-1) + (n-2)*T(n-1, n-k) for 1 < k <= n, T(n, 1) = T(n-1, n-1) for n > 0, T(n, 0) = 0^n.

1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 2, 6, 8, 8, 0, 8, 32, 50, 56, 56, 0, 56, 280, 480, 608, 640, 640, 0, 640, 3840, 6880, 9280, 10680, 10960, 10960, 0, 10960, 76720, 140800, 196480, 237760, 260800, 264640, 264640, 0, 264640, 2117120, 3942720, 5607040, 6982400, 7968000, 8505040, 8581760, 8581760

Offset: 0

Author

Mikhail Kurkov, Jul 19 2025

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  0,     1;
  0,     1,     1;
  0,     1,     2,      2;
  0,     2,     6,      8,      8;
  0,     8,    32,     50,     56,     56;
  0,    56,   280,    480,    608,    640,    640;
  0,   640,  3840,   6880,   9280,  10680,  10960,  10960;
  0, 10960, 76720, 140800, 196480, 237760, 260800, 264640, 264640;
		

Links

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

Crossrefs

Cf. A008281, A386381 (main diagonal).

Programs

Maple

T := proc(n, k) option remember; ifelse(k = 0, 0^n, ifelse(k = 1, T(n-1, n-1), T(n, k-1) + (n - 2)*T(n-1, n-k))) end: seq(seq(T(n, k), k = 0..n), n = 0..9); # Peter Luschny, Jul 19 2025

Mathematica

A386363[n_, k_] := A386363[n, k] = Switch[k, 0, Boole[n == 0], 1, A386363[n-1, n-1], _, A386363[n, k-1] + (n-2)*A386363[n-1, n-k]];
Table[A386363[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jul 21 2025 *)

PARI

rows_upto(n) = {my(v1, v2, v3);
v1 = vector(n+1, i, 0); v1[1] = 1;
v2 = vector(n+1, i, 0); v2[1] = [1];
for(i=1, n, v3 = v1; v1[1] = 0; v1[2] = v3[i];
for(j=2, i, v1[j+1] = v1[j] + (i-2)*v3[i-j+1]);
v2[i+1] = vector(i+1, j, v1[j])); v2}

Python

from functools import cache
@cache
def seidel(n: int)-> list[int]:
    if n == 0: return [1]
    rowA = seidel(n - 1)
    row = [0] + seidel(n - 1)
    row[1] = row[n]
    for k in range(2, n + 1):
        row[k] = row[k - 1] + (n - 2) * rowA[n - k]
    return row
def A386363row(n: int) -> list[int]: return seidel(n)
for n in range(10): print(A386363row(n))  # Peter Luschny, Jul 20 2025