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A026012
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Second differences of Catalan numbers A000108.
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9
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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
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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Table of n, a(n) for n=0..26.
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FORMULA
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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 DELEHAM, 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
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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T(2n, n), where T is the array defined in A026009.
Cf. A127158.
Sequence in context: A094831 A033193 A071738 * A191993 A120900 A059712
Adjacent sequences: A026009 A026010 A026011 * A026013 A026014 A026015
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Clark Kimberling
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STATUS
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approved
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