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A083886
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E.g.f. exp(3*x)*exp(x^2).
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2
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1, 3, 11, 45, 201, 963, 4899, 26253, 147345, 862083, 5238459, 32957037, 214117209, 1433320515, 9867008979, 69734001357, 505212273441, 3747124863747, 28418591888235, 220152270759597, 1740363304031721, 14027180742479043, 115176800996769411, 962726355659386125, 8186311912829551281, 70769800810139187843
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OFFSET
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0,2
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COMMENTS
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Binomial transform of A000898.
Hankel transform is A108400. [From Paul Barry, Jun 13 2009]
a(n) is the number of self-inverse signed permutations of length 2n that are equal to their reverse-complements and avoid the pattern (-2,-1). As a result, a(n) also gives the same thing but for avoiding any one of (-1,-2), (+2,+1) or (+1,+2) instead of (-2,-1). (See the Hardt and Troyka reference.) - Justin M. Troyka, Aug 05 2011.
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REFERENCES
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A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, pre-print. [For more information, email troykaj(AT)carleton.edu.]
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LINKS
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Table of n, a(n) for n=0..25.
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FORMULA
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E.g.f.: exp(3*x+x^2).
From Paul Barry, Jun 13 2009: (Start)
G.f.: 1/(1-3x-2x^2/(1-3x-4x^2/(1-3x-6x^2/(1-3x-8x^2/(1-... (continued fraction);
a(n) = sum{k=0..floor(n/2), C(n,2k)(2k)!3^(n-2k)/k!}. (End)
a(n) = i^n*Hermite_H(n, -3i/2), i=sqrt(-1). [Paul Barry, Jun 15 2009]
a(0) = 1; a(1) = 3; a(n) = 3*a(n-1) + 2*(n-1)*a(n-2) for n >= 2. - Justin M. Troyka, Aug 05 2011.
E.g.f. 1 + (x+3)*x/(G(0)-x^2-3*x) where G(k)= x^2 + 3*x + k + 1 - (x+3)*x*(k+1)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Jul 12 2012
G.f.: 1/Q(0) where Q(k) = 1 + 2*x*k - 2*x - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013
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EXAMPLE
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Since a(2) = 11, there are 11 self-inverse signed permutations of 4 that are equal to their reverse-complements and avoid (-2,-1). Some of these are: (+3,+4,+1,+2), (+4,-2,-3,+1), (-1,+3,+2,-4), (-1,-2,-3,-4). - Justin M. Troyka, Aug 05 2011.
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MATHEMATICA
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a = {1, 3}; For[n = 2, n < 13, n++, a = Append[a, 3 a[[n]] + 2 (n - 1) a[[n - 1]]]]; a // Justin M. Troyka, Aug 5 2011.
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PROG
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(PARI) x='x+O('x^66); Vec(serlaplace(exp(3*x)*exp(x^2))) /* Joerg Arndt, Jul 12 2012 */
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CROSSREFS
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Cf. A047974, A001813.
Sequence in context: A187764 A151133 A213333 * A030866 A030941 A030918
Adjacent sequences: A083883 A083884 A083885 * A083887 A083888 A083889
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry, May 09 2003
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STATUS
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approved
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