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A026012 Second differences of Catalan numbers A000108. 11
1, 2, 6, 19, 62, 207, 704, 2431, 8502, 30056, 107236, 385662, 1396652, 5088865, 18642420, 68624295, 253706790, 941630580, 3507232740, 13105289370, 49114150020, 184560753390, 695267483664, 2625197720454, 9933364416572, 37660791173152, 143048202990504 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = s(2n) = 2.

Number of Dyck paths of semilength n+2 with no initial and no final UD's. Example: a(2)=6 because the only Dyck paths of semilength 4 with no initial and no final UD's are: UUDUDUDD, UUDUUDDD, UUUDDUDD, UUUDUDDD, UUDDUUDD, UUUUDDDD. - Emeric Deutsch, Oct 26 2003

Number of branches of length 1 starting from the root in all ordered trees with n+1 edges. Example: a(1)=2 because the tree /\ has two branches of length 1 starting from the root and the path-tree of length 2 has none. a(n)=Sum(k*A127158(n+1,k),k=0..n+1). - Emeric Deutsch, Mar 01 2007

Number of staircase walks from (0,0) to (n,n) that never cross y=x+2. Example: a(3) = 19 because up,up,up,right,right,right is not allowed but the other binomial(6,3)-1 = 19 paths are. - Mark Spindler, Nov 11 2012

REFERENCES

S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see pp. 188, 196).

Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

LINKS

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

FORMULA

Expansion of (1+x^1*C^3)*C^1, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

a(n) = 3*(3*n^2+3*n+2)*binomial(2*n, n)/((n+1)*(n+2)*(n+3)) - Emeric Deutsch, Oct 26 2003

a(n) = Sum_{k, 0<=k<=2} A039599(n,k) = A000108(n)+A000245(n)+A000344(n). [From Philippe Deléham, Nov 12 2008]

a(n) = binomial(2*n,n)/(n+1)*hypergeom([-2,n+1/2],[n+2],4). - Peter Luschny, Aug 15 2012

a(n) = binomial(2*n,n)-binomial(2n,n-3) - Mark Spindler, Nov 11 2012

Conjecture: (n+3)*a(n) +(-5*n-6)*a(n-1) +2*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jun 20 2013

MATHEMATICA

Differences[Table[CatalanNumber[n], {n, 0, 28}], 2] (* Jean-François Alcover, Sep 28 2012 *)

Table[Binomial[2n, n]-Binomial[2n, n-3], {n, 0, 26}] (* Mark Spindler, Nov 11 2012 *)

PROG

(PARI) a(n) = 3*(3*n^2+3*n+2)*binomial(2*n, n)/((n+1)*(n+2)*(n+3)); /* Joerg Arndt, Aug 19 2012 */

CROSSREFS

T(2n, n), where T is the array defined in A026009.

Cf. A127158, A059346.

Sequence in context: A094831 A033193 A071738 * A191993 A120900 A059712

Adjacent sequences:  A026009 A026010 A026011 * A026013 A026014 A026015

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Clark Kimberling

STATUS

approved

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Last modified September 17 21:45 EDT 2014. Contains 246885 sequences.