A080337 Bisection of A080107.

1, 3, 12, 59, 339, 2210, 16033, 127643, 1103372, 10269643, 102225363, 1082190554, 12126858113, 143268057587, 1778283994284, 23120054355195, 314017850216371, 4444972514600178, 65435496909148513, 999907522895563403, 15832873029742458796, 259377550023571768075

Offset: 1

Author

Wouter Meeussen, Mar 18 2003

Keywords

• nonn

Comments

Number of symmetric positions of non-attacking rooks on upper-diagonal part of 2n X 2n chessboard.

Number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k)<=2+max(prefix) for k>=1, see example. - Joerg Arndt, Apr 25 2010

Number of achiral color patterns in a row or loop of length 2n-1. Two color patterns are equivalent if the colors are permuted. - Robert A. Russell, Apr 24 2018

Stirling transform of A005425(n-1) per Knuth reference. - Robert A. Russell, Apr 28 2018

Examples

			From _Joerg Arndt_, Apr 25 2010: (Start)
For n=0 there is one empty string (term a(0)=0 not included here); for n=1 there is one string [0]; for n=2 there are 3 strings [00], [01], and [02];
for n=3 there are a(3)=12 strings (in lexicographic order):
01: [000],
02: [001],
03: [002],
04: [010],
05: [011],
06: [012],
07: [013],
08: [020],
09: [021],
10: [022],
11: [023],
12: [024].
(End)
For a(3) = 12, both the row and loop patterns are AAAAA, AABAA, ABABA, ABBBA, AABCC, ABACA, ABBBC, ABCAB, ABCBA, ABCBD, ABCDA, and ABCDE. - _Robert A. Russell_, Apr 24 2018
		

References

• D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 765). - Robert A. Russell, Apr 28 2018

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..514
• Joerg Arndt, Matters Computational (The Fxtbook), section 17.3.4, pp. 364-366.
• Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 18, 29.
• J. Quaintance, Letter representations of rectangular m x n x p proper arrays, arXiv:math/0412244 [math.CO], 2004-2006.

Crossrefs

Row sums of A140735.

Cf. A002872, A080107.

Column k=2 of A305962.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j)), j=1..m+2))
    end:
a:= n-> b(n, -1):
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 15 2018

Mathematica

Table[Sum[ Binomial[n, k] A002872[[k + 1]], {k, 0, n}], {n, 0, 24}]
Aodd[m_, k_] := Aodd[m, k] = If[m > 1, k Aodd[m-1, k] + Aodd[m-1, k-1]
  + Aodd[m-1, k-2], Boole[m==1 && k==1]]
Table[Sum[Aodd[m, k], {k, 1, 2m-1}], {m, 1, 30}] (* Robert A. Russell, Apr 24 2018 *)
x[n_] := x[n] = If[n<2, n+1, 2x[n-1] + (n-1) x[n-2]]; (* A005425 *)
Table[Sum[StirlingS2[n, k] x[k-1], {k, 0, n}], {n, 30}] (* Robert A. Russell, Apr 28 2018, after Knuth reference *)

PARI

x='x+O('x^66);
egf=exp(x+exp(x)+exp(2*x)/2-3/2); /* = 1 +3*x +6*x^2 +59/6*x^3 +113/8*x^4 +... */
Vec(serlaplace(egf)) /* Joerg Arndt, Apr 29 2011 */

Formula

Binomial transform of A002872 (sorting numbers).

E.g.f.: exp(x+exp(x)+exp(2*x)/2-3/2) = exp(x+sum(j=1,2, (exp(j*x)-1)/j ) ). - Joerg Arndt, Apr 29 2011

From Robert A. Russell, Apr 24 2018: (Start)

Aodd[n,k] = [n>1]*(k*Aodd[n-1,k]+Aodd[n-1,k-1]+Aodd[n-1,k-2])+[n==1]*[k==1]

a(n) = Sum_{k=1..2n-1} Aodd[n,k]. (End)

a(n) = Sum_{k=0..n} Stirling2(n, k)*A005425(k-1). (from Knuth reference) - Robert A. Russell, Apr 28 2018

Extensions

Comment corrected by Wouter Meeussen, Aug 14 2009