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A262233 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=1 if n>0.

%I A262233 #18 Jan 02 2021 07:59:57
%S A262233 1,1,1,3,20,220,3648,84616,2617696,104112576,5176135040,314525766016,
%T A262233 22934467613184,1976385358538240,198701625441195520,
%U A262233 23050434113386398720,3055967615464202301440,459172688072604359835648,77616824553405653653094400
%N A262233 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=1 if n>0.
%H A262233 Vaclav Kotesovec, <a href="/A262233/b262233.txt">Table of n, a(n) for n = 0..250</a> (terms 0..110 from Alois P. Heinz)
%F A262233 a(n) ~ c * d^n * n!^2 / n, where d = 0.552406011965766199179395470003589240257321... and c = 1.48557711044485933585341072480938... - _Vaclav Kotesovec_, Sep 18 2020
%e A262233 a(1) = 1: (1,1).
%e A262233 a(2) = 1: (12,12).
%e A262233 a(3) = 3: (123,123), (132,132), (132,231).
%e A262233 a(4) = 20: (1234,1234), (1243,1243), (1243,1342), (1243,2341), (1324,1324), (1324,1423), (1324,2314), (1324,2413), (1324,3412), (1342,1243), (1342,1342), (1342,2341), (1423,1324), (1423,1423), (1423,2314), (1423,2413), (1423,3412), (1432,1432), (1432,2431), (1432,3421).
%p A262233 b:= proc(u, o, h) option remember; `if`(u+o=0, 1,
%p A262233       add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+
%p A262233       add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))
%p A262233     end:
%p A262233 a:= n-> `if`(n=0, 1, add(b(j-1, n-j, n-1), j=1..n)):
%p A262233 seq(a(n), n=0..20);
%t A262233 b[u_, o_, h_] := b[u, o, h] = If[u + o == 0, 1,
%t A262233    Sum[Sum[b[u - j, o + j - 1, h + i - 1], {i, 1, u + o - h}], {j, 1, u}]+
%t A262233    Sum[Sum[b[u + j - 1, o - j, h - i], {i, 1, h}], {j, 1, o}]];
%t A262233 a[n_] := If[n == 0, 1, Sum[b[j - 1, n - j, n - 1], {j, 1, n}]];
%t A262233 a /@ Range[0, 20] (* _Jean-François Alcover_, Jan 02 2021, after _Alois P. Heinz_ *)
%Y A262233 Cf. A060350, A262234, A262241.
%K A262233 nonn
%O A262233 0,4
%A A262233 _Alois P. Heinz_, Sep 15 2015