cp's OEIS Frontend

A223094 Number of foldings of n labeled stamps in which leaf n is inwards.

%I A223094 #42 Jul 24 2025 05:10:44
%S A223094 0,0,2,6,26,78,288,888,3130,9850,34112,108998,374636,1211046,4148816,
%T A223094 13533796,46304730,152153758,520434552,1720325302,5885686496,
%U A223094 19552190624,66927118548,223264746520,764725528072,2560239468774,8775478294368,29470844083770
%N A223094 Number of foldings of n labeled stamps in which leaf n is inwards.
%C A223094 Subset of foldings of n labeled stamps (A000136). [_Stéphane Legendre_, Apr 09 2013]
%C A223094 From _Roger Ford_, Aug 23 2024: (Start)
%C A223094 a(n) represents the number of impossible stamp foldings with stamp 1 on top and n+1 stamps that are correctly folded for the first n stamps. From stamp n to stamp n+1, the stamp connection crosses a folding so the folding is impossible.
%C A223094 Example a(3) = 2.  Impossible foldings = 1,3,2,4 and 1,4,2,3.
%C A223094                1 ____                      1 ____
%C A223094 Stamp numbers  3 ____|__   Vertical Lines  4 ____|__
%C A223094                2 |___|  |  lines are folds 2 ____| |
%C A223094                4 _______|                  3 |_____|
%C A223094 a(4) = 6, and that means for 5 stamps there are 6 impossible foldings with the first impossible folding occurring from stamp 4 to stamp 5.  Impossible foldings = 1,2,4,3,5; 1,2,5,3,4; 1,3,4,2,5; 1,4,3,5,2; 1,5,2,4,3; 1,5,3,4,2. (End)
%H A223094 Stéphane Legendre, <a href="/A223094/b223094.txt">Table of n, a(n) for n = 1..43</a>
%H A223094 Stéphane Legendre, <a href="http://arxiv.org/abs/1302.2025">Foldings and Meanders</a>, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
%H A223094 <a href="/index/Fo#fold">Index entries for sequences obtained by enumerating foldings</a>
%F A223094 a(n) = A000136(n) - A000682(n+1). - _Andrew Howroyd_, Dec 05 2015
%F A223094 For n >= 3: a(n) = n! - Sum_{k=3..n-1} (a(k)*n!/k!) - A000682(n+1). - _Roger Ford_, Aug 24 2024
%t A223094 A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];
%t A223094 a[n_] := n A000682[[n]] - A000682[[n + 1]];
%t A223094 Array[a, 43] (* _Jean-François Alcover_, Sep 02 2019 *)
%Y A223094 Cf. A000136, A000682.
%K A223094 nonn
%O A223094 1,3
%A A223094 _N. J. A. Sloane_, Mar 29 2013
%E A223094 More terms from _Stéphane Legendre_, Apr 09 2013