0,3

Also the dimension of SL(3)-invariants in V^n tensor (V^*)^n, where V is the standard 3-dimensional representation of SL(3) and V^* is its dual. - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

Also the number of doubly-alternating permutations of length 2n with no four-term increasing subsequence (i.e., 1234-avoiding doubly-alternating permutations). The doubly-alternating permutations (counted by sequence A007999) are those permutations w such that both w and w^(-1) have descent set {2, 4, 6, ...}. [From Joel Brewster Lewis (jblewis(AT)post.harvard.edu), May 21 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Gessel, Ira M., Symmetric functions and P-recursiveness. J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.

O. Guibert, E. Pergola, Enumeration of vexillary involutions which are equal to their mirror/complement, Discrete Math., Vol. 224 (2000), pp. 281-287.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.16(e), p. 453.

R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull. Amer. Math. Soc., 40 (2003), 55-68.

J. Noonan and D. Zeilberger, [math/9808080] The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns

J. Noonan and D. Zeilberger, The Enumeration of Permutations With A Prescribed Number of "Forbidden" Patterns

R. P. Stanley, A combinatorial miscellany

Erik Ouchterlony, Pattern avoiding doubly alternating permutations [From Joel Brewster Lewis (jblewis(AT)post.harvard.edu), May 21 2009]

a(n) = 2 * sum_{k=0..n} binomial(2*k, k) * (binomial(n, k))^2 * (3*k^2+2*k+1-n-2*k*n)/((k+1)^2 * (k+2) * (n-k+1))

(4*n^2 - 2*n + 1)*(n + 2)^2*(n + 1)^2*a(n) = (44*n^3 - 14*n^2 - 11*n + 8)*n*(n + 1)^2*a(n - 1) - (76*n^4 + 42*n^3 - 49*n^2 - 24*n + 24)*(n - 1)^2*a(n - 2) + 9*(4*n^2 + 6*n + 3)*(n - 1)^2*(n - 2)^2*a(n - 3). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 16 2004

a(0) = 1, a(1) = 1, (n^2 + 8 n + 16) a(n + 2) = (10 n^2 + 42 n + 41) a(n + 1) - (9 n^2 + 18 n + 9) a(n) - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

a := n->2*sum(binomial(2*k, k)*(binomial(n, k))^2*(3*k^2+2*k+1-n-2*k*n)/ (k+1)^2/(k+2)/(n-k+1), k=0..n);

A005802:=rsolve({a(0) = 1, a(1) = 1, (n^2 + 8*n + 16)*a(n + 2) = (10*n^2 + 42*n + 41)*a(n + 1) - (9*n^2 + 18*n + 9)*a(n)}, a(n), makeproc): (Mihailovs)

a[n_] := 2Sum[Binomial[2k, k]Binomial[n, k]^2(3k^2+2k+1-n-2k*n)/((k+1)^2(k+2)(n-k+1)), {k, 0, n}]

A column of A047888.

Sequence in context: A088929 A004040 A022558 this_sequence A061552 A053488 A117106

Adjacent sequences: A005799 A005800 A005801 this_sequence A005803 A005804 A005805

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)

Additional comments from Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 06 2000

More terms from Naohiro Nomoto (6284968128(AT)geocities.co.jp), Jun 18 2001

Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Dec 10 2002

More terms from Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005