A370767 - OEIS

%I #11 Mar 06 2024 17:59:02

%S 1,1,3,17,139,1401,16867,236513,3787707,68219081,1364931859,

%T 30037136433,721044433387,18750182814233,525071095004739,

%U 15753703863875201,504159100060894747,17142539126080474473,617165134818228049267,23453349764127439545041

%N Number of signed permutations of length n+1 with adjacent elements differing by more than 1 and whose first element is 1.

%C A signed permutation is a sequence (x_1,x_2,...,x_n) of integers such that {|x_1|,|x_2|,...|x_n|} = {1,2...,n}.

%C Adjacent elements that differ in sign will always differ by more than 1.

%H Andrew Howroyd, <a href="/A370767/b370767.txt">Table of n, a(n) for n = 0..200</a>

%F A283184(n) = a(n) - a(n-1) for n > 0.

%F a(n) = (1+2*n)*a(n-1) + (3-2*n)*a(n-2) + (5-2*n)*a(n-3) + (-4+2*n)*a(n-4) for n >= 4.

%e In the following examples, the number of assignments of signs to each unsigned permutation is shown in parenthesis.

%e a(2) = 3: 123(1), 132(2). Total is 1 + 2 = 3.

%e a(3) = 17: 1234(1), 1243(2), 1324(4), 1342(4), 1423(4), 1432(2).

%o (PARI) a(n)=subst(serlaplace(polcoef(1/(1 + (1 - 2*y)*x + 2*y*x^2) + O(x*x^n), n)), y, 1)

%Y Cf. A283184, A370766, A370768.

%K nonn

%O 0,3

%A _Andrew Howroyd_, Mar 01 2024