A177322 - OEIS

%I #13 Nov 06 2024 04:41:31

%S 12,660,51240,4635540,457507512,47768769048,5188083048720,

%T 580132098966420,66341857216154520,7722843117550721160,

%U 912113857017595941072,109025503164832356811800

%N Number of permutations of n copies of 1..4 with all adjacent differences <= 2 in absolute value.

%H R. H. Hardin, <a href="/A177322/b177322.txt">Table of n, a(n) for n=1..35</a>

%F From _Peter Bala_, Nov 05 2024: (Start)

%F The following are conjectural:

%F For n >= 1, a(n) = Sum_{k = 0..2*n} (-1)^(n+k) * (k/n)^2 * binomial(2*n, k)^4. Cf. the identity Sum_{k = 0..2*n} (-1)^(n+k) * (k/n) * binomial(2*n, k)^2 = binomial(2*n, n) = A000984(n) for n >= 1.

%F For n >= 1, a(n) = 2 * binomial(2*n, n) * Sum_{k = 0..n} (k/n) * binomial(2*n, n-k)^2 * binomial(2*n+k, k).

%F P-recursive: n^3*(2*n-1)*(n-1)*(24*n^3-105*n^2+152*n-73)*a(n) = 2*(n-1)*(3264*n^7-20808*n^6+53900*n^5-73159*n^4+55963*n^3-24107*n^2+5436*n-504)*a(n-1) - 4*(2*n-1)*(24*n^3-33*n^2+14*n-2)*(2*n-3)^2*(n-2)^2*a(n-2) with a(1) = 12 and a(2) = 660.

%F The supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r.(End)

%Y Cf. A050983. A177316 - A177328.

%K nonn

%O 1,1

%A _R. H. Hardin_, May 06 2010