0,2

Counts binary rooted trees (with out-degree <=2), embedded in plane, with n labeled end nodes of degree 1. Unlabeled version gives Catalan numbers A000108.

Define a "downgrade" to be the permutation which places the items of a permutation in descending order. We are concerned with permutations that are identical to their downgrades. Only permutations of order 4n and 4n+1 can have this property; the number of permutations of length 4n having this property are equinumerous with those of length 4n+1. If a permutation p has this property then the reversal of this permutation also has it. a(n) = number of permutations of length 4n and 4n+1 that are identical to their downgrades. - Eugene McDonnell (eemcd(AT)mac.com), Oct 26 2003

a(n)=12*A051618(a) n>=2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 15 2008

P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D 14 (1976), 1536-1553.

D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6, Eq. 32.

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.

McDonnell, Eugene, "Magic Squares and Permutations" APL Quote-Quad 7.3 (Fall, 1976)

R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).

H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.

N. J. A. Sloane, Table of n, a(n) for n = 0..100

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

W. Y. C. Chen, L. W. Shapiro and L. L. M. Young, Parity reversing involutions on plane trees and 2-Motzkin paths

P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D14 (1976), 1536-1553.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 115

E. Lucas, Th\'{e}orie des Nombres. Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.

R. J. Marsh and P. P. Martin, Pascal arrays: counting Catalan sets

C. Radoux, Determinants de Hankel et theoreme de Sylvester

Index entries for related partition-counting sequences

A. F. Labossiere, Sobalian Coefficients.

A. F. Labossiere, Miscellaneous.

E.g.f. (1-4*x)^(-1/2). a(n) = (2*n)!/n! = product[ k=0..n-1 ] (4*k+2).

Integral representation as n-th moment of a positive function on a positive half-axis, in Maple notation: a(n)=int(x^n*exp(-x/4)/(sqrt(x)*2*sqrt(Pi)), x=0..infinity), n=0, 1, .. . This representation is unique. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 18 2001

Define a'(1)=1, a'(n)=sum(k=1, n-1, a'(n-k)*a'(k)*C(n, k)); then a(n)=a'(n+1) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 27 2003

With interpolated zeros (1, 0, 2, 0, 12, ...) this has e.g.f. exp(x^2). - Paul Barry (pbarry(AT)wit.ie), May 09 2003

a(n) = A000680(n)/A000142(n)*A000079(n) = product(4*i+2, i=0..n-1) = 4^n*pochhammer(1/2, n) = 4^n*GAMMA(n+1/2)/sqrt(Pi) - Daniel Dockery (peritus(AT)gmail.com) Jun 13, 2003

For asymptotics see the Robinson paper.

a(k) = (2*k)!/k! = Sum_{i=1..k+1} |A008275(i,k+1)| * k^(i-1) - Andre F. Labossiere (boronali(AT)laposte.net), Jun 21 2007

a(n)=A000984(n)*A000142(n) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 25 2008

The following permutations of order 8 and their reversals have this property:

1 7 3 5 2 4 0 6

1 7 4 2 5 3 0 6

2 3 7 6 1 0 4 5

2 4 7 1 6 0 3 5

3 2 6 7 0 1 5 4

3 5 1 7 0 6 2 4

A001813 := n->(2*n)!/n!;

spec := [ B, {B=Union(Z, Prod(B, B))}, labeled ]; [seq(combstruct[count](spec, size=n), n=1..20)];

For Maple program see A000903.

BB:=[T, {T=Prod(Z, F), F=Sequence(B), B=Prod(F, Z)}, labeled]: seq(count(BB, size=i), i=1..17); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2007

seq(mul((n+k), k=1..n), n=0..16); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 15 2008

(Mupad) combinat::catalan(n)*(n+1)*n! $ n = 0..16 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 15 2007

Cf. A037224, A048854, A001147, A007696, A008545, A122670 (essentially the same sequence).

Cf. A000165, A001813, A047055, A047657, A084947, A084948, A084949.

Cf. A010050, A000142, A008275, A000108, A000984, A008276, A000680, A094216.

Sequence in context: A096317 A081470 A108135 this_sequence A097388 A131815 A047793

Adjacent sequences: A001810 A001811 A001812 this_sequence A001814 A001815 A001816

nonn,easy,nice

njas

More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 01 2000