

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
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OFFSET

0,2


COMMENTS

Number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and s(i)  s(i1) = 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 pathtree 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
Number of standard Young tableaux of skew shape (n+2,n)/(2), for n>=2.  Ran Pan, Apr 07 2015


REFERENCES

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


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.


FORMULA

Expansion of (1+x^1*C^3)*C^1, where C = (1(14*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).  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,n3).  Mark Spindler, Nov 11 2012
Conjecture: (n+3)*a(n) + (5*n6)*a(n1) + 2*(2*n3)*a(n2) = 0.  R. J. Mathar, Jun 20 2013
E.g.f.: exp(2*x)*(BesselI(0,2*x)  BesselI(3,2*x)).  Ilya Gutkovskiy, Feb 28 2017


MATHEMATICA

Differences[Table[CatalanNumber[n], {n, 0, 28}], 2] (* JeanFrançois Alcover, Sep 28 2012 *)
Table[Binomial[2n, n]Binomial[2n, n3], {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 A284216
Adjacent sequences: A026009 A026010 A026011 * A026013 A026014 A026015


KEYWORD

nonn,changed


AUTHOR

N. J. A. Sloane, Clark Kimberling


STATUS

approved



