A386381 Main diagonal of A386363.

1, 1, 1, 2, 8, 56, 640, 10960, 264640, 8581760, 360331520, 19031302400, 1235451750400, 96722377139200, 8988790940876800, 978442125179648000, 123324448870740377600, 17820979140159760793600, 2926936219425738642227200, 542215853077506417192140800, 112527512540808439576566169600

Offset: 0

Author

Mikhail Kurkov, Jul 20 2025

Keywords

• nonn
• changed

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..264

Crossrefs

Cf. A000010, A000111, A060818, A386363.

Programs

Maple

a := proc(n) local T; 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: T(n, n) end:
seq(a(n), n = 0..20);  # Peter Luschny, Jul 21 2025

PARI

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] = v1[i+1]); v2}

Formula

This sequence has surprising divisibility properties. Let E(n) = A000111(n) and phi(n) = A000010(n).

Conjecture 1: if d is a divisor of a(k), then d is also divisor of a(m*d + k) where m is any natural number. In particular, a(k) is a divisor of a(m*a(k) + k) for any k >= 0 where m is any natural number.

Conjecture 2: if d > 2 is a divisor of E(k), then d is also a divisor of E(m*phi(d)+k) where m is any natural number. In particular, E(k) is a divisor of E(m*phi(E(k)) + k) for any k >= 0 (with single exception at k = 3) where m is any natural number.

From Peter Luschny, Jul 20 2025: (Start)

Conjecture: 2*a(n) is divisible by A060818(n). (End)

a(n) ~ c * 2^(n+1) * n^(2*n-3) / (exp(2*n) * Pi^(n-1)), where c = 25.574519628957467521537232312735336894... - Vaclav Kotesovec, Sep 02 2025