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A065096
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Sums of lists produced by a variant of the iteration that produces the Catalan numbers: start with 0 and at each iteration replace each integer k by the list 0,1,...,k-1,k,k+1,k,k-1,...,1,0 and let a(n) be the sum of the resulting (flattened) list after n iterations.
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3
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0, 1, 6, 31, 156, 785, 3978, 20335, 104856, 545073, 2854350, 15046383, 79787700, 425360481, 2278586898, 12259138975, 66216193968, 358941938849, 1952111592342, 10648449309823, 58245727453260, 319406931168241, 1755674399021466, 9671384910586511, 53384080026230856, 295225111836281425, 1635532359053982558, 9075703373174900943, 50439671788908739428, 280733665349833191873, 1564624146618843908130, 8731422123788687832639
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Number of diagonals emanating from a fixed vertex of a convex (n+3)-gon in all of its dissections. Example: a(1)=1 because in the three dissections of a convex quadrilateral ABCD (namely: empty, {AC}, {BD}) there is only one diagonal emanating from A.
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FORMULA
| G.f.=(1-3z-sqrt(1-6z+z^2))^2/(16z^3).
a(n)=(1/pi)*Int(x^n*sqrt(-x^2+6x-1)*(x-3)/8,x,3-2sqrt(2),3+2sqrt(2)); - Paul Barry (pbarry(AT)wit.ie), Sep 16 2006
a(0) = 0 and, for n>0, a(n) = Sum_{k=1..n} A001003(k)*A001003(n+1-k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 27 2004
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MATHEMATICA
| Table[Plus@@Flatten[Nest[ #/.a_Integer:> Join[Range[0, a+1], Range[a, 0, -1]]&, {0}, n]], {n, 0, 10}]
Table[Range[n, 0, -1].Table[a[n, k], {k, 0, n}], {n, 0, 36}] with a[n, k] as defined in A033877.
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CROSSREFS
| Cf. A000108, A001003.
Sequence in context: A026705 A003463 A026771 * A077352 A038223 A022034
Adjacent sequences: A065093 A065094 A065095 * A065097 A065098 A065099
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KEYWORD
| nonn
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AUTHOR
| Wouter Meeussen (wouter.meeussen(AT)pandora.be), Nov 11 2001
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