A382910 a(n) = A003266(n)^2.

1, 1, 1, 4, 36, 900, 57600, 9734400, 4292870400, 4962558182400, 15011738501760000, 118907980672440960000, 2465675887223735746560000, 133859078241489389944995840000, 19025256931384645503492313743360000, 7079298104168226591849489943904256000000, 6896432754839457130755425769163265163264000000

Offset: 0

Author

Edwin Hermann, Apr 08 2025

Keywords

• nonn
• easy

Comments

For n>=3 number of valid symmetrical change ringing methods on n bells with the shortest number of rows per lead where the treble plain hunts out to the back. See Wikipedia and the Polster and Ross link for an explanation of bell ringing terminology.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..70
• Richard Duckworth and Fabian Stedman, Tintinnalogia, or, the Art of Ringing, (1671). Released by Project Gutenberg, 2006.
• Burkard Polster and Marty Ross, Ringing the changes, (2009).
• Wikipedia, Method ringing.
• Index entries for sequences related to bell ringing.

Crossrefs

Cf. A000045, A001622, A003266, A007598, A062073, A090281.

Programs

Maple

a:= proc(n) a(n):= `if`(n=0, 1, a(n-1)*(<<0|1>, <1|1>>^n)[1, 2]^2) end:
seq(a(n), n=0..16);  # Alois P. Heinz, Apr 14 2025

Mathematica

k = 1; {1, 1}~Join~Reap[Do[k *= Fibonacci[n]; Sow[k^2], {n, 16}] ][[-1, 1]] (* Michael De Vlieger, Apr 14 2025 *)

Formula

a(n) = Product_{j=1..n} Fibonacci(j)^2.

a(0) = 1; a(n) = a(n-1)*A007598(n). - Hugo Pfoertner, Apr 13 2025

a(n) ~ c^2 * phi^(n*(n+1)) / 5^n where phi is the golden ratio (A001622) and c = A062073. - Amiram Eldar, Aug 18 2025