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%I M1786 N0705 #121 Aug 31 2023 10:52:36
%S 1,2,7,31,164,999,6841,51790,428131,3827967,36738144,376118747,
%T 4086419601,46910207114,566845074703,7186474088735,95318816501420,
%U 1319330556537631,19013488408858761,284724852032757686,4422344774431494155,71125541977466879231
%N Number of partitions of {1..2n} that are invariant under a permutation consisting of n 2-cycles.
%C Previous name was: Sorting numbers.
%C a(n) = number of symmetric partitions of the set {-n,...,-1,1,...,n}. A partition of {-n,...,-1,1,...,n} into nonempty subsets X_1,...,X_k is 'symmetric' if for each i, -X_i=X_j for some j. a(n) = S_B(n,1)+...+S_B(n,n) where S_B(n,k) is as in A085483. a(n) is the n-th Bell number of 'type B'. - _James East_, Aug 18 2003
%C Column 2 of A162663. - _Franklin T. Adams-Watters_, Jul 09 2009
%C a(n) is equal to the sum of all expressions of the form p(1^n)[st(lambda)] for partitions lambda of order less than or equal to n, where p(1^n)[st(lambda)] denotes the coefficient of the irreducible character basis element indexed by the partition lambda in the expansion of the power sum basis element indexed by the partition (1^n). - _John M. Campbell_, Sep 16 2017
%C Number of achiral color patterns in a row or loop of length 2n. Two color patterns are equivalent if the colors are permuted. - _Robert A. Russell_, Apr 24 2018
%C Stirling transform of A005425 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A002872/b002872.txt">Table of n, a(n) for n = 0..513</a> (first 101 terms from T. D. Noe)
%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. 12, 18, 29.
%H T. Halverson and M. Reeks, <a href="https://arxiv.org/abs/1302.6150">Gelfand Models for Diagram Algebras</a>, arXiv preprint arXiv:1302.6150 [math.RT], 2013.
%H Vaclav Kotesovec, <a href="https://arxiv.org/abs/2207.10568">Asymptotics for a certain group of exponential generating functions</a>, arXiv:2207.10568 [math.CO], Jul 13 2022.
%H Victor Meally, <a href="/A002868/a002868.pdf">Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.</a>
%H T. S. Motzkin, <a href="/A000262/a000262.pdf">Sorting numbers for cylinders and other classification numbers</a>, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
%H J. Pasukonis and S. Ramgoolam, <a href="https://arxiv.org/abs/1010.1683">From counting to construction for BPS states in N=4SYM</a>, arXiv:1010.1683 [hep-th], 2010, (E.2).
%H J. Pasukonis and S. Ramgoolam, <a href="https://doi.org/10.1007/JHEP02(2011)078">From counting to construction for BPS states in N=4SYM</a>, J. High En. Phys. 2011 (2) (2011), (E.2).
%H OEIS Wiki, <a href="http://oeis.org/wiki/Sorting_numbers">Sorting numbers</a>
%H R. Orellana and M. Zabrocki, <a href="https://arxiv.org/abs/1605.06672">Symmetric group characters as symmetric functions</a>, arXiv preprint arXiv:1605.06672 [math.CO], 2016-2017.
%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.
%H <a href="/index/So#sorting">Index entries for sequences related to sorting</a>
%F E.g.f.: e^( (e^(2x) - 3)/2 + e^x ).
%F a(n) = A080107(2n) for all n. - _Jörgen Backelin_, Jan 13 2016
%F From _Robert A. Russell_, Apr 24 2018: (Start)
%F Aeven(n,k) = [n>0]*(k*Aeven(n-1,k)+Aeven(n-1,k-1)+Aeven(n-1,k-2))
%F + [n==0]*[k==0]
%F a(n) = Sum_{k=0..2n} Aeven(n,k). (End)
%F a(n) = Sum_{k=0..n} Stirling2(n, k)*A005425(k). (from Knuth reference) - _Robert A. Russell_, Apr 28 2018
%F a(n) ~ exp(exp(2*r)/2 + exp(r) - 3/2 - n) * (n/r)^(n + 1/2) / sqrt((1 + 2*r)*exp(2*r) + (1 + r)*exp(r)), where r = LambertW(2*n)/2 - 1/(1 + 2/LambertW(2*n) + n^(1/2) * (1 + LambertW(2*n)) * (2/LambertW(2*n))^(3/2)). - _Vaclav Kotesovec_, Jul 03 2022
%F a(n) ~ (2*n/LambertW(2*n))^n * exp(n/LambertW(2*n) + (2*n/LambertW(2*n))^(1/2) - n - 7/4) / sqrt(1 + LambertW(2*n)). - _Vaclav Kotesovec_, Jul 10 2022
%e For a(2)=7, the row patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD. The loop patterns are AAAA, AAAB, AABB, AABC, ABAB, ABAC, and ABCD. - _Robert A. Russell_, Apr 24 2018
%p a:= proc(n) option remember; `if`(n=0, 1, add((1+
%p 2^(j-1))*binomial(n-1, j-1)*a(n-j), j=1..n))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Oct 29 2015
%t u[0,j_]:=1;u[k_,j_]:=u[k,j]=Sum[Binomial[k-1,i-1]Plus@@(u[k-i,j]#^(i-1)&/@Divisors[j]),{i,k}]; Table[u[n,2],{n,0,12}] (* _Wouter Meeussen_, Dec 06 2008 *)
%t mx = 16; p = 2; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* _Robert G. Wilson v_, Dec 12 2012 *)
%t Aeven[m_, k_] := Aeven[m, k] = If[m>0, k Aeven[m-1, k] + Aeven[m-1, k-1]
%t + Aeven[m-1, k-2], Boole[m==0 && k==0]]
%t Table[Sum[Aeven[m, k], {k, 0, 2m}], {m, 0, 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], {k, 0, n}], {n, 0, 20}] (* _Robert A. Russell_, Apr 28 2018, from Knuth reference *)
%t Table[Sum[Binomial[n,k] * 2^k * BellB[k, 1/2] * BellB[n-k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jun 29 2022 *)
%Y Cf. A002873, A002874, A080107, A085483.
%Y u[n,j] is A162663.
%Y Row sums of A293181.
%Y Column k=2 of A306024.
%Y Cf. A005425.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
%E Edited by _Franklin T. Adams-Watters_, Jul 09 2009
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