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

0,2

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

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

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 *)

Crossrefs

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

Sequence in context: A367386 A166296 A151527 * A152172 A001677 A339150

Adjacent sequences: A086622 A086623 A086624 * A086626 A086627 A086628

Keyword

nonn

Author

Paul D. Hanna, Jul 24 2003

Status

approved