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A145885 a(n) = (n-1)^2*binomial(2n,n)/[2(n+1)]. 1
0, 1, 10, 63, 336, 1650, 7722, 35035, 155584, 680238, 2939300, 12584726, 53488800, 225990180, 950094810, 3977737875, 16594533120, 69018792150, 286296636780, 1184823735810, 4893253404000, 20171905282620, 83020426503300 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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, followed 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.

REFERENCES

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

LINKS

Table of n, a(n) for n=1..23.

FORMULA

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.

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

(n+1)*(n-2)^2*a(n) -2*(2*n-1)*(n-1)^2*a(n-1)=0. - R. J. Mathar, Aug 10 2017

MAPLE

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

MATHEMATICA

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

CROSSREFS

Cf. A000108, A002740, A002054, A145884.

Sequence in context: A055368 A278802 A077616 * A093953 A298067 A298716

Adjacent sequences:  A145882 A145883 A145884 * A145886 A145887 A145888

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 06 2008

STATUS

approved

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Last modified June 16 10:03 EDT 2019. Contains 324152 sequences. (Running on oeis4.)