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A177322 Number of permutations of n copies of 1..4 with all adjacent differences <= 2 in absolute value.

%I A177322 #24 Jun 07 2025 15:34:11
%S A177322 1,12,660,51240,4635540,457507512,47768769048,5188083048720,
%T A177322 580132098966420,66341857216154520,7722843117550721160,
%U A177322 912113857017595941072,109025503164832356811800,13164173606420256001705200,1603262885152270822600633200,196721396289915224779758846240
%N A177322 Number of permutations of n copies of 1..4 with all adjacent differences <= 2 in absolute value.
%H A177322 Alois P. Heinz, <a href="/A177322/b177322.txt">Table of n, a(n) for n = 0..471</a> (terms n=1..35 from R. H. Hardin)
%F A177322 From _Peter Bala_, Nov 05 2024: (Start)
%F A177322 The following are conjectural:
%F A177322 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 A177322 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 A177322 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(0) = 1.
%F A177322 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)
%F A177322 a(n) = A156554(n) * A000984(n). A156554 counts ways to place 1s and 4s in the permutation; a positive (resp. negative) sequence element in A156554 is a run-length of 1s (resp. 4s) followed by a 2 or 3 or the end of the permutation. Each zero in A156554 corresponds to an additional 2 or 3 in the permutation. - _Martin Fuller_, Jun 07 2025
%Y A177322 Cf. A050983. A177316 - A177328, A156554.
%K A177322 nonn
%O A177322 0,2
%A A177322 _R. H. Hardin_, May 06 2010
%E A177322 a(0)=1 prepended by _Alois P. Heinz_, Jun 07 2025