%I #65 Aug 31 2023 10:52:23
%S 1,3,12,59,339,2210,16033,127643,1103372,10269643,102225363,
%T 1082190554,12126858113,143268057587,1778283994284,23120054355195,
%U 314017850216371,4444972514600178,65435496909148513,999907522895563403,15832873029742458796,259377550023571768075
%N Bisection of A080107.
%C Number of symmetric positions of non-attacking rooks on upper-diagonal part of 2n X 2n chessboard.
%C 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
%C 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
%C Stirling transform of A005425(n-1) per Knuth reference. - _Robert A. Russell_, Apr 28 2018
%D 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
%H Alois P. Heinz, <a href="/A080337/b080337.txt">Table of n, a(n) for n = 1..514</a>
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 17.3.4, pp. 364-366.
%H Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, <a href="https://arxiv.org/abs/2308.14183">Combinatorial Identities for Vacillating Tableaux</a>, arXiv:2308.14183 [math.CO], 2023. See pp. 18, 29.
%H J. Quaintance, <a href="https://arxiv.org/abs/math/0412244">Letter representations of rectangular m x n x p proper arrays</a>, arXiv:math/0412244 [math.CO], 2004-2006.
%F Binomial transform of A002872 (sorting numbers).
%F 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
%F From _Robert A. Russell_, Apr 24 2018: (Start)
%F Aodd[n,k] = [n>1]*(k*Aodd[n-1,k]+Aodd[n-1,k-1]+Aodd[n-1,k-2])+[n==1]*[k==1]
%F a(n) = Sum_{k=1..2n-1} Aodd[n,k]. (End)
%F a(n) = Sum_{k=0..n} Stirling2(n, k)*A005425(k-1). (from Knuth reference) - _Robert A. Russell_, Apr 28 2018
%e From _Joerg Arndt_, Apr 25 2010: (Start)
%e 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];
%e for n=3 there are a(3)=12 strings (in lexicographic order):
%e 01: [000],
%e 02: [001],
%e 03: [002],
%e 04: [010],
%e 05: [011],
%e 06: [012],
%e 07: [013],
%e 08: [020],
%e 09: [021],
%e 10: [022],
%e 11: [023],
%e 12: [024].
%e (End)
%e 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
%p b:= proc(n, m) option remember; `if`(n=0, 1,
%p add(b(n-1, max(m, j)), j=1..m+2))
%p end:
%p a:= n-> b(n, -1):
%p seq(a(n), n=1..25); # _Alois P. Heinz_, Jun 15 2018
%t Table[Sum[ Binomial[n, k] A002872[[k + 1]], {k, 0, n}], {n, 0, 24}]
%t Aodd[m_, k_] := Aodd[m, k] = If[m > 1, k Aodd[m-1, k] + Aodd[m-1, k-1]
%t + Aodd[m-1, k-2], Boole[m==1 && k==1]]
%t Table[Sum[Aodd[m, k], {k, 1, 2m-1}], {m, 1, 30}] (* _Robert A. Russell_, Apr 24 2018 *)
%t x[n_] := x[n] = If[n<2, n+1, 2x[n-1] + (n-1) x[n-2]]; (* A005425 *)
%t Table[Sum[StirlingS2[n, k] x[k-1], {k, 0, n}], {n, 30}] (* _Robert A. Russell_, Apr 28 2018, after Knuth reference *)
%o (PARI) x='x+O('x^66);
%o 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 +... */
%o Vec(serlaplace(egf)) /* _Joerg Arndt_, Apr 29 2011 */
%Y Row sums of A140735.
%Y Cf. A002872, A080107.
%Y Column k=2 of A305962.
%K nonn
%O 1,2
%A _Wouter Meeussen_, Mar 18 2003
%E Comment corrected by _Wouter Meeussen_, Aug 14 2009