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

0, 0, 2, 6, 26, 78, 288, 888, 3130, 9850, 34112, 108998, 374636, 1211046, 4148816, 13533796, 46304730, 152153758, 520434552, 1720325302, 5885686496, 19552190624, 66927118548, 223264746520, 764725528072, 2560239468774, 8775478294368, 29470844083770

Offset: 1

Author

N. J. A. Sloane, Mar 29 2013

Keywords

• nonn

Comments

Subset of foldings of n labeled stamps (A000136). [Stéphane Legendre, Apr 09 2013]

From Roger Ford, Aug 23 2024: (Start)

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.

Example a(3) = 2. Impossible foldings = 1,3,2,4 and 1,4,2,3.

1 __ 1 __

Stamp numbers 3 __|__ Vertical Lines 4 __|__

2 |_| | lines are folds 2 __| |

4 _____| 3 |___|

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)

Links

• Stéphane Legendre, Table of n, a(n) for n = 1..43
• Stéphane Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
• Index entries for sequences obtained by enumerating foldings

Crossrefs

Cf. A000136, A000682.

Programs

Mathematica

A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];
a[n_] := n A000682[[n]] - A000682[[n + 1]];
Array[a, 43] (* Jean-François Alcover, Sep 02 2019 *)

Formula

a(n) = A000136(n) - A000682(n+1). - Andrew Howroyd, Dec 05 2015

For n >= 3: a(n) = n! - Sum_{k=3..n-1} (a(k)*n!/k!) - A000682(n+1). - Roger Ford, Aug 24 2024

Extensions

More terms from Stéphane Legendre, Apr 09 2013