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A226435
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Number of permutations of 1..n with fewer than 2 interior elements having values lying between the values of their neighbors.
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1
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1, 2, 6, 22, 90, 422, 2226, 13102, 85170, 606542, 4697946, 39330982, 353985450, 3408792662, 34975509666, 380947661662, 4390028664930, 53368010874782, 682564606249386, 9162253729773142, 128794752680027610, 1892150024227428902
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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E.g.f. (for offset 0, conjecture): (sec(x) + tan(x)) - (sec(x) + tan(x))^2 + (sec(x) + tan(x))^3. - Sergei N. Gladkovskii, Jun 11 2015
a(n) = Sum_{i=0..(n-2)/2}((n-2*i-1)*Sum_{j=0..2*i}((-1)^(j+i)*2^(-n-j+2*i+2)*Stirling2(n,n+j-2*i)*binomial(n+j-2*i-1,n-2*i-1)*(n+j-2*i)!)), n > 1, a(1)=1. - Vladimir Kruchinin, Apr 08 2016
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EXAMPLE
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Some solutions for n=9:
..1...9...4...3...2...6...1...4...2...7...3...3...2...6...5....6
..7...2...7...1...5...3...7...1...5...2...8...1...3...1...4....3
..2...3...5...6...6...9...2...6...3...8...6...9...1...4...8....8
..4...1...6...4...3...1...5...5...1...6...9...4...5...3...1....1
..9...6...1...7...7...7...3...8...9...5...1...8...4...5...2....7
..5...5...9...9...4...5...9...3...6...9...5...5...9...7...9....4
..8...7...2...2...8...8...8...2...7...1...2...2...6...2...3....5
..3...4...8...8...1...4...4...9...4...4...7...7...8...9...7....9
..6...8...3...5...9...2...6...7...8...3...4...6...7...8...6....2
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MATHEMATICA
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CoefficientList[Series[Sec[x]+Tan[x] - (Sec[x]+Tan[x])^2 + (Sec[x]+Tan[x])^3, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 11 2015 after Sergei N. Gladkovskii, all 210 terms match those in the b-file *)
{1}~Join~Table[Sum[(n - 2 i - 1) Sum[(-1)^(j + i)*2^(-n - j + 2 i + 2) StirlingS2[n, n + j - 2 i] Binomial[n + j - 2 i - 1, n - 2 i - 1] (n + j - 2 i)!, {j, 0, 2 i}], {i, 0, (n - 2)/2}], {n, 2, 22}] (* Michael De Vlieger, Apr 08 2016 *)
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PROG
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(Maxima) a(n):=sum((n-2*i-1)*sum((-1)^(j+i)*2^(-n-j+2*i+2)*stirling2(n, n+j-2*i)*binomial(n+j-2*i-1, n-2*i-1)*(n+j-2*i)!, j, 0, 2*i), i, 0, (n-2)/2); /* Vladimir Kruchinin, Apr 08 2016 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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