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

Hankel transform is A137565. [From Paul Barry (pbarry(AT)wit.ie), Nov 25 2009]

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

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

A. F. Labossiere, Sobalian Coefficients.

A. F. Labossiere, Miscellaneous.

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

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 127

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

Let b(n)=b(n-1)+4; then a(n)=b(n)*a(n-1). - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Sep 17 2008

Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 15 2009: (Start)

G.f.: 1/(1-2x/(1-4x/(1-6x/(1-8x/(1-10x/(1-...... (continued fraction);

a(n)=(n+1)!*A000108(n); (End)

a(n)=sum{k=0..n, A132393(n,k)*2^(2n-k)}. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 10 2009]

G.f.: 1/(1-2x-8x^2/(1-10x-48x^2/(1-18x-120x^2/(1-26x-224x^2/(1-34x-360x^2/(1-42x-528x^2/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Nov 25 2009]

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

k = 4; b[1] = 2; b[n_] := b[n] = b[n - 1] + k; a[0] = 1; a[1] = 2; a[n_] := a[n] = a[n - 1]*b[n]; Table[a[n], {n, 0, 20}] - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Sep 17 2008

s=1; lst={s}; Do[s+=n*s; AppendTo[lst, s], {n, 1, 5!, 4}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 08 2008]

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

(Other) sage: [binomial(2*n, n)*factorial(n) for n in xrange(0, 17)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 03 2009]

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

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

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

Sequence in context: A096317 A081470 A108135 this_sequence A097388 A152188 A131815

Adjacent sequences: A001810 A001811 A001812 this_sequence A001814 A001815 A001816

nonn,easy,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

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