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A002872
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Number of partitions of {1..2n} that are invariant under a permutation consisting of n 2-cycles.
(Formerly M1786 N0705)
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26
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1, 2, 7, 31, 164, 999, 6841, 51790, 428131, 3827967, 36738144, 376118747, 4086419601, 46910207114, 566845074703, 7186474088735, 95318816501420, 1319330556537631, 19013488408858761, 284724852032757686, 4422344774431494155, 71125541977466879231
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OFFSET
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0,2
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COMMENTS
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Previous name was: Sorting numbers.
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
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
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
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REFERENCES
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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
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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E.g.f.: e^( (e^(2x) - 3)/2 + e^x ).
Aeven(n,k) = [n>0]*(k*Aeven(n-1,k)+Aeven(n-1,k-1)+Aeven(n-1,k-2))
+ [n==0]*[k==0]
a(n) = Sum_{k=0..2n} Aeven(n,k). (End)
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
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
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EXAMPLE
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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
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MAPLE
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a:= proc(n) option remember; `if`(n=0, 1, add((1+
2^(j-1))*binomial(n-1, j-1)*a(n-j), j=1..n))
end:
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MATHEMATICA
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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 *)
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 *)
Aeven[m_, k_] := Aeven[m, k] = If[m>0, k Aeven[m-1, k] + Aeven[m-1, k-1]
+ Aeven[m-1, k-2], Boole[m==0 && k==0]]
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], {k, 0, n}], {n, 0, 20}] (* Robert A. Russell, Apr 28 2018, from Knuth reference *)
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 *)
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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