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A386381 Main diagonal of A386363.

%I A386381 #20 Sep 03 2025 05:16:57
%S A386381 1,1,1,2,8,56,640,10960,264640,8581760,360331520,19031302400,
%T A386381 1235451750400,96722377139200,8988790940876800,978442125179648000,
%U A386381 123324448870740377600,17820979140159760793600,2926936219425738642227200,542215853077506417192140800,112527512540808439576566169600
%N A386381 Main diagonal of A386363.
%H A386381 Vaclav Kotesovec, <a href="/A386381/b386381.txt">Table of n, a(n) for n = 0..264</a>
%F A386381 This sequence has surprising divisibility properties. Let E(n) = A000111(n) and phi(n) = A000010(n).
%F A386381 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.
%F A386381 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.
%F A386381 From _Peter Luschny_, Jul 20 2025: (Start)
%F A386381 Conjecture: 2*a(n) is divisible by A060818(n). (End)
%F A386381 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
%p A386381 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:
%p A386381 seq(a(n), n = 0..20);  # _Peter Luschny_, Jul 21 2025
%o A386381 (PARI) upto(n) = {my(v1, v2, v3);
%o A386381 v1 = vector(n+1, i, 0); v1[1] = 1;
%o A386381 v2 = vector(n+1, i, 0); v2[1] = 1;
%o A386381 for(i=1, n, v3 = v1; v1[1] = 0; v1[2] = v3[i];
%o A386381 for(j=2, i, v1[j+1] = v1[j] + (i-2)*v3[i-j+1]);
%o A386381 v2[i+1] = v1[i+1]); v2}
%Y A386381 Cf. A000010, A000111, A060818, A386363.
%K A386381 nonn,changed
%O A386381 0,4
%A A386381 _Mikhail Kurkov_, Jul 20 2025