OFFSET
1,3
COMMENTS
a(n) is the subset of the set of unordered pairings of the first 2n integers (A001147) forbidding pairs of the form (i,i+1) for all i in [2,n-1]. There are many other selections of forbidden pairs giving the same count. - Olivier Gérard, Feb 08 2011
REFERENCES
G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 1..201
Simon Plouffe, Approximations of generating functions and a few conjectures, arXiv:0911.4975 [math.NT], 2009.
FORMULA
a(n) = |A000806(n-1)|+|A000806(n)|. G.f.: Sum_{n>=0} A001147(n)*(x/(1+x)^2)^n. - Vladeta Jovovic, Jun 27 2007
Recurrence: (4*n^2-8*n+1)*a(n-1) + (2*n-1)*a(n-2) + (3-2*n)*a(n) = 0. - Vaclav Kotesovec, Oct 05 2012
G.f.: T(0) - 1, where T(k) = 1 - (k+1)*x/( (k+1)*x - (1+x)^2/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 03 2013
a(-n) = -a(n) for all n in Z. - Michael Somos, Jan 27 2014
a(n+1) = Sum_{k=0..n} (-1)^k * (2n+1-k)! / (2^(n-k) * k! * (n-k)!) if n>=0. - Michael Somos, Jan 27 2014
0 = a(n) * (a(n+2) + a(n+3)) + a(n+1) * (-a(n+1) -3*a(n+2) -4*a(n+3) + a(n+4)) + a(n+2) * (-3*a(n+3) + a(n+4)) + a(n+3) * (-a(n+3)) for all n in Z. - Michael Somos, Jan 27 2014
E.g.f. (for offset 0): ((2 - 2*x - (1 - 2*x)^(1/2)) / (1-2*x)^(3/2)) * exp((1-2*x)^(1/2) - 1) (formula due to B. Salvy, see Plouffe link). - Gheorghe Coserea, Aug 05 2015
E.g.f. (for offset 1): exp(sqrt(1-2*x)-1) * (1/sqrt(1-2*x)-1). - Vaclav Kotesovec, Nov 29 2015
a(n) ~ 2^(n+1/2)*n^n/exp(n+1). - Vaclav Kotesovec, Nov 29 2015
EXAMPLE
G.f. = x + x^2 + 6*x^3 + 41*x^4 + 365*x^5 + 3984*x^6 + 51499*x^7 + ...
MATHEMATICA
a[ n_] := With[ {m = Abs[n] - 1}, If[ m < 0, 0, Sign[n] Hypergeometric1F1[-m, -2 m - 1, -2] (2 m + 1)!!]]; (* Michael Somos, Jan 27 2014 *)
a[ n_] := With[ {m = Abs[n] - 1}, If[ m < 0, 0, Sign[n] Sum[ (-1)^k (2 m + 1 - k)! / (2^(m - k) k! (m - k)!), {k, 0, m}]]]; (* Michael Somos, Jan 27 2014 *)
a[ n_] := With[ {m = Abs[n] - 1}, If[ m < 0, 0, Sign[n] Numerator @ FromContinuedFraction[ Table[(-1)^Quotient[k, 2] If[ OddQ[k], k, 1], {k, 2 m + 1}]]]]; (* Michael Somos, Jan 27 2014 *)
Rest[CoefficientList[Series[E^(-1 + Sqrt[1 - 2*x])*(-1 + 1/Sqrt[1 - 2*x]), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Nov 29 2015 *)
Table[(2 n - 1)!! Hypergeometric1F1[1 - n, 1 - 2 n, -2], {n, 20}] (* Eric W. Weisstein, Nov 14 2018 *)
PROG
(PARI) {a(n) = sign(n) * if( n==0, 0, contfracpnqn( vector( 2*abs(n) -1, k, (-1)^(k\2) * if( k%2, k, 1))) [1, 1]) }; /* Michael Somos, Jan 27 2014 */
(PARI) {a(n) = sign(n) * sum( k=0, n=abs(n)-1, (-1)^k * (2*n + 1 - k)! / (2^(n - k) * k! * (n - k)!) ) }; /* Michael Somos, Jan 27 2014 */
(PARI) x = 'x+O('x^33); Vec(serlaplace(((2 - 2*x - (1 - 2*x)^(1/2)) / (1-2*x)^(3/2)) * exp((1-2*x)^(1/2) - 1))) \\ Gheorghe Coserea, Aug 05 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved