A086625 Antidiagonal sums of square table A086623.

1, 2, 3, 6, 12, 26, 59, 138, 332, 814, 2028, 5118, 13054, 33598, 87143, 227542, 597640, 1577866, 4185108, 11146570, 29798682, 79932298, 215072896, 580327122, 1569942098, 4257254850, 11569980794, 31508150890, 85968266198, 234975421554

Offset: 0

Author

Paul D. Hanna, Jul 24 2003

Keywords

• nonn

Comments

a(n) is the number of Dyck (n+1)-paths (A000108) containing no DUDD and no UUPDD where P is a nonempty Dyck subpath. Example: a(2)=3 counts UUDDUD, UDUUDD, UDUDUD but omits UUUDDD because it contains an offending UUPDD and omits UUDUDD because it contains a DUDD. - David Callan, Oct 26 2006

Links

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

Crossrefs

Cf. A086623 (table), A086624 (diagonal).

Programs

Mathematica

CoefficientList[Series[(-1+x+Sqrt[1+x*(-2-3*x+4*x^3)])/(2*(-1+x)*x^2),{x,0,20}],x] (* Vaclav Kotesovec, Mar 22 2014 *)

Formula

G.f.: A(x) = (1-x^2)/(1-x)^2 + x^2*A(x)^2.

a(n) ~ sqrt(11*r-3) / (4*sqrt(2*Pi)*(1-r)*n^(3/2)*r^(n+5/2)), where r = 0.3478103847799310287... is the root of the equation 4*r^3+4*r^2+r = 1. - Vaclav Kotesovec, Mar 22 2014

D-finite with recurrence (n+2)*a(n) +2*(-n-1)*a(n-1) +(-3*n+4)*a(n-2) +4*a(n-3) +4*(n-3)*a(n-4)=0. - R. J. Mathar, Sep 29 2020