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A289489 Number of permutations p of [n] such that in 0p the sum of all jumps equals 2n.

%I A289489 #17 Nov 17 2022 04:06:32
%S A289489 1,0,0,1,4,15,104,644,3696,23388,151842,979110,6445659,43148963,
%T A289489 290832906,1977914328,13574296048,93787977144,651970844448,
%U A289489 4558718881927,32038664402074,226200869873851,1603811085640698,11415385190127413,81538284501095235
%N A289489 Number of permutations p of [n] such that in 0p the sum of all jumps equals 2n.
%C A289489 An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.
%H A289489 Alois P. Heinz, <a href="/A289489/b289489.txt">Table of n, a(n) for n = 0..300</a>
%F A289489 a(n) = A291722(n,n).
%F A289489 a(n) ~ c * d^n / n^2, where d = 7.7572369635460295... and c = 0.022080578979754... - _Vaclav Kotesovec_, Nov 17 2022
%e A289489 a(3) = 1: 312.
%e A289489 a(4) = 4: 3142, 4213, 4231, 4312.
%e A289489 a(5) = 15: 15234, 25134, 31542, 35124, 41235, 42153, 42531, 43152, 45123, 53214, 53241, 53421, 54213, 54231, 54312.
%e A289489 a(6) = 104: 126354, 136254, 142635, 146253, ..., 653421, 654213, 654231, 654312.
%p A289489 b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
%p A289489       add(b(u-j, o+j-1)*x^(j-1), j=1..u)+
%p A289489       add(b(u+j-1, o-j)*x^(j-1), j=1..o)))
%p A289489     end:
%p A289489 a:= n-> coeff(b(0, n), x, n):
%p A289489 seq(a(n), n=0..26);
%t A289489 b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1,
%t A289489    Sum[b[u - j, o + j - 1]*x^(j - 1), {j, 1, u}] +
%t A289489    Sum[b[u + j - 1, o - j]*x^(j - 1), {j, 1, o}]]];
%t A289489 a[n_] := Coefficient[b[0, n], x, n];
%t A289489 Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Nov 17 2022, after _Alois P. Heinz_ *)
%Y A289489 Cf. A291722.
%K A289489 nonn
%O A289489 0,5
%A A289489 _Alois P. Heinz_, Sep 02 2017