1,3

a(n)=sum of valley abscissae in all Dyck paths of semilength n minus number of valleys in all Dyck paths of semilength (n). Example: a(3)=10; indeed, the Dyck paths of semilength 3, fiollowed by their valley abscissae are UDUDUD (2,4), UDUUDD (2), UUDDUD (4), UUDUDD (3), UUUDDD ( ); therefore a(3)=2+4+2+4+3 - 5 = 10. Instead of Dyck paths one can consider Dyck words; then sum of valley abscissae corresponds to major index and number of valleys to number of descents.

a(n)=A002740(n+1)-A002054(n-1) (n>=2).

a(n)=Sum(k*A145884(n,k),k=0..(n-1)^2) for n>=1.

a(n)=(n-1)^2*Cat(n)/2, where Cat(n)=binomial(2n,n)/(n+1)=A000108(n) are the Catalan numbers.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see p. 236, Exercise 6.34 d.

G.f.=4z^2[8z-1+3sqrt(1-4z)]/[(1+sqrt(1-4z))^3*(1-4z)^(3/2)].

seq((1/2)*(n-1)^2*binomial(2*n, n)/(n+1), n=1..24);

Table[CatalanNumber[n]*(n - 1)^2/2, {n, 1, 23}] [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 08 2009]

A000108, A002740, A002054, A145884

Sequence in context: A159240 A055368 A077616 this_sequence A093953 A075755 A046638

Adjacent sequences: A145882 A145883 A145884 this_sequence A145886 A145887 A145888

nonn

Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 06 2008