0,3

The number of rooted Eulerian n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets (liskov(AT)im.bas-net.by), Mar 17 2005

This is also the number of strings of length 2n of two different types of balanced parentheses. For example, a(2) = 4, since the two possible strings of length 4 are [()] and ([]). - Jeffrey Shallit (shallit(AT)graceland.uwaterloo.ca), Jun 03 2006

Row sums of number triangle A110506. - Paul Barry (pbarry(AT)wit.ie), Jul 24 2005

Also row sums of triangle in A085880 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 01 2005

Row sums of number triangle A114608. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 15 2008]

V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.

M. Bousquet-Melou, Limit laws for embedded trees

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 651

V. A. Liskovets and T. R. Walsh, Counting unrooted maps on the plane, Advances in Applied Math., 36, No.4 (2006), 364-387.

8^(n-1)*GAMMA(n-1/2)/GAMMA(n+1)/Pi^(1/2), n>0.

Recurrence: {a(1)=1, (-4+8*n)*a(n)-(n+1)*a(n+1)}

G.f.: (1-sqrt(1-8x))/4 = xC(2x) where C(x) is g.f. for Catalan numbers, A000108.

G.f. A(x) satisfies 2A(x)^2-A(x)+x=0, A(0)=0 and A(x)=x+2A(x)^2=x/(1-2A(x)). - Michael Somos, Sep 06 2003

a(0)=0, a(1)=1; a(n)=2*sum(i=1, n-1, a(i)*a(n-i)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 16 2004

With a different offset, a(0)=1, a(n)=sum{k=0..n, sum{j=0..n, j*C(2n-j-1, n-j)C(j, k)2^(n-j)/n}}, n>0 - Paul Barry (pbarry(AT)wit.ie), Jul 24 2005

With a different offset, a(n) = 2^n*A000108(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 31 2005

The Hankel transform of a(n+1)=[1,2,8,40,224,1344,...] is 4^C(n+1,2) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 06 2007

spec := [S, {B=Union(C, Z), S=Union(B, C), C=Prod(S, S)}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

InverseSeries[Series[y-2*y^2], {y, 0, 24}], x] (* then A(x)=y(x) *) - Len Smiley Apr 07 2000

(PARI) a(n)=if(n<1, 0, 2^(n-1)*(2*n-2)!/(n-1)!/n!)

(PARI) a(n)=if(n<1, 0, polcoeff(serreverse(x-2*x^2+x*O(x^n)), n))

(PARI) a(n)=if(n<1, 0, polcoeff(2*x/(1+sqrt(1-8*x+O(x^n))), n))

a(n)=A052714(n)/n!. a(n)=A003645(n-2)*2, n>1.

Limit of array A102544.

Cf. A003645.

Sequence in context: A006195 A092807 A074601 this_sequence A151374 A085485 A089603

Adjacent sequences: A052698 A052699 A052700 this_sequence A052702 A052703 A052704

easy,nonn

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Better description from Claude Lenormand (claude.lenormand(AT)free.fr), Mar 19 2001

Additional comments from Michael Somos, Feb 24, 2002