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A144828
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Partial products of successive terms of A017113; a(0)=1.
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15
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1, 4, 48, 960, 26880, 967680, 42577920, 2214051840, 132843110400, 9033331507200, 686533194547200, 57668788341964800, 5305528527460761600, 530552852746076160000, 57299708096576225280000, 6646766139202842132480000, 824199001261152424427520000, 108794268166472120024432640000
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of signed permutations of length 4n that are equal to their reverse-inverses. Note that the reverse-inverse of a permutation is equivalent to a 90-degree rotation of the permutation's diagram (see the Hardt and Troyka link). - Justin M. Troyka, Aug 11 2011
Define the bar operation as an operation on signed permutation that flips the sign of each entry. Then a(n) is the number of signed permutations of length 2n that are equal to the bar of their inverses and equal to their reverse-complements (see the Hardt and Troyka link). - Justin M. Troyka, Aug 11 2011
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LINKS
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A. Hardt and J. M. Troyka, Slides (associated with the Hardt and Troyka reference above).
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FORMULA
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a(n) = Sum_{k=0..n} A132393(n,k)*4^k*8^(n-k).
G.f.: 1/(1-4x/(1-8x/(1-12x/(1-16x/(1-20x/(1-24x/(1-28x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
a(n) = (-4)^n*Sum_{k=0..n} 2^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) = 8^n * Gamma(n + 1/2) / sqrt(Pi). - Daniel Suteu, Jan 06 2017
0 = a(n)*(8*a(n+1) - a(n+2)) + a(n+1)*(+a(n+1)) and a(n) = (-1)^n / a(-n) for all n in Z. - Michael Somos, Jan 06 2017
Sum_{n>=0} 1/a(n) = 1 + e^(1/8)*sqrt(Pi)*erf(1/(2*sqrt(2)))/(2*sqrt(2)), where erf is the error function. - Amiram Eldar, Dec 20 2022
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EXAMPLE
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a(0)=1, a(1)=4, a(2)=4*12=48, a(3)=4*12*20=960, a(4)=4*12*20*28=26880, ...
Since a(1) = 4, there are 4 signed permutations of 4 that are equal to their reverse-inverses. These are: (+2,+4,+1,+3), (+3,+1,+4,+2), (-2,-4,-1,-3), (-3,-1,-4,-2). - Justin M. Troyka, Aug 11 2011
G.f. = 1 + 4*x + 48*x^2 + 960*x^3 + 26880*x^4 + 967680*x^5 + 42577920*x^6 + ...
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MAPLE
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MATHEMATICA
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Join[{1}, FoldList[Times, (8*Range[0, 20]+4)]] (* Harvey P. Dale, Dec 01 2015 *)
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PROG
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(Magma) [2^k *Factorial(2*k) / Factorial(k): k in [0..20]]; // Vincenzo Librandi, Aug 11 2011
(PARI) {a(n) = if( n<0, (-1)^n / a(-n), 2^n *(2*n)! / n!)}; /* Michael Somos, Jan 06 2017 */
(Sage) [2^n*factorial(n+1)*catalan_number(n) for n in (0..30)] # G. C. Greubel, Apr 02 2021
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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