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A125107
Subtract compositions (A011782) from Catalan numbers (A000108).
2
0, 0, 0, 1, 6, 26, 100, 365, 1302, 4606, 16284, 57762, 205964, 738804, 2666248, 9678461, 35324902, 129579254, 477507628, 1767001046, 6563596132, 24465218444, 91480466488, 343055419346, 1289895758716, 4861929624236, 18367319517720
OFFSET
0,5
COMMENTS
Apparently the number of Dyck n-paths with more than half of the path lying between the first and last peaks. - David Scambler, Sep 14 2012
FORMULA
a(n) = A000108(n) - A011782(n).
Conjecture: (n+1)*a(n) +2*(1-4*n)*a(n-1) +4*(5*n-7)*a(n-2) +8*(5-2*n)*a(n-3)=0. - R. J. Mathar, Aug 10 2013
EXAMPLE
A000108 begins 1 1 2 5 14 42 132 429 ...
A011782 begins 1 1 2 4 8 16 32 64 ...
so we get .... 0 0 0 1 6 26 100 365 ...
MATHEMATICA
Table[CatalanNumber[n] - If[n==0, 1, 2^(n-1)], {n, 0, 30}] (* David Scambler, Sep 14 2012 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Alford Arnold, Dec 15 2006
EXTENSIONS
More terms from David Scambler, Sep 14 2012
STATUS
approved