%I #21 Apr 21 2022 05:25:09

%S 1,1,9,156,3098,69274,1626122,39892080,1004867492,25886899652,

%T 677767802220,17984050212906,482214668573802,13042214648300918,

%U 355247290177412584,9733704443846822462,268026951144933433138,7411550898419782031320,205686202884689885529314

%N Number of permutations p of [2n] such that in 0p the largest up-jump equals n and no down-jump is larger than 2.

%H Alois P. Heinz, <a href="/A320290/b320290.txt">Table of n, a(n) for n = 0..250</a>

%F a(n) = A291680(2n,n).

%p b:= proc(u, o, k) option remember; `if`(u+o=0, 1,

%p add(b(u-j, o+j-1, k), j=1..min(2, u))+

%p add(b(u+j-1, o-j, k), j=1..min(k, o)))

%p end:

%p a:= n-> `if`(n=0, 1, b(0, 2*n, n)-b(0, 2*n, n-1)):

%p seq(a(n), n=0..20);

%t b[u_, o_, k_] := b[u, o, k] = If[u + o == 0, 1,

%t Sum[b[u - j, o + j - 1, k], {j, 1, Min[2, u]}] +

%t Sum[b[u + j - 1, o - j, k], {j, 1, Min[k, o]}]];

%t a[n_] := If[n == 0, 1, b[0, 2*n, n] - b[0, 2*n, n - 1]];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Apr 21 2022, after _Alois P. Heinz_ *)

%Y Cf. A291680.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Oct 27 2018