A086405 Row T(n,3) of number array A086404.

1, 4, 18, 84, 396, 1872, 8856, 41904, 198288, 938304, 4440096, 21010752, 99423936, 470479104, 2226331008, 10535111424, 49852682496, 235905426432, 1116316463616, 5282466223104, 24996898556928, 118286594002944

Offset

0,2

Comments

Binomial transform of A079935.

Number of nonisomorphic graded posets with 0 of rank n+1, with exactly 2 elements of each rank level above 0. Here, we do not assume all maximal elements have maximal rank and thus use graded poset to mean: For every element x, all maximal chains among those with x as greatest element have the same finite length. - David Nacin, Feb 13 2012

References

R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Wikipedia, Graded Poset

Index entries for linear recurrences with constant coefficients, signature (6,-6).

Formula

G.f.: (1-2*x)/((1-(3-sqrt(3))*x)*(1-(3+sqrt(3))*x)) = (1-2*x)/(1-6*x+6*x^2);

a(n) = (3-sqrt(3))^n*(1/2 - 1/(2*sqrt(3))) + (3 + sqrt(3))^n*(1/2 + 1/(2*sqrt(3))).

E.g.f.: exp(3*x)*(cosh(sqrt(3*x) + sinh(sqrt(3)*x)/sqrt(3)). - Paul Barry, Nov 20 2003

a(n) = Sum_{k=1..floor(n/2)} C(n, 2k)*3^(n-k-1). - Paul Barry, Nov 22 2003

a(n) = (((1+sqrt(3))*(3+sqrt(3))^n) - ((1-sqrt(3))*(3-sqrt(3))^n))/sqrt(12). - Al Hakanson (hawkuu(AT)gmail.com), Jun 10 2009

a(n) = Sum_{k=0..n} A117317(n,k)*2^k. - Philippe Deléham, Jan 28 2012

a(n) = 6*(a(n-1) - a(n-2)), a(0)=1, a(1)=4. - David Nacin, Feb 27 2012

G.f.: (1-2*x)/(1-6*x+6*x^2). - Colin Barker, Aug 04 2012

Mathematica

LinearRecurrence[{6, -6}, {1, 4}, 60] (* David Nacin, Feb 27 2012 *)

Prog

(Python)
def a(n, adict={0:1, 1:4}):
    if n in adict:
        return adict[n]
    adict[n]=6*a(n-1)-6*a(n-2)
    return adict[n] # David Nacin, Feb 27 2012

Crossrefs

Cf. A079935, A086404.

Sequence in context: A290916 A014348 A126020 * A151251 A293490 A010849

Adjacent sequences: A086402 A086403 A086404 * A086406 A086407 A086408

Keyword

easy,nonn

Author

Paul Barry, Jul 19 2003

Status

approved