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A058987
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a(n) = Catalan(n) - Motzkin(n-1).
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3
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0, 1, 3, 10, 33, 111, 378, 1303, 4539, 15961, 56598, 202214, 727389, 2632605, 9581211, 35047098, 128791323, 475281921, 1760726808, 6545921136, 24415415001, 91340016081, 342658850427, 1288774386909, 4858753673655, 18358309669651
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OFFSET
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1,3
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COMMENTS
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Number of Dyck paths with a "small Capital N" (a rise then a fall then a rise) - this follows from the exercise on p. 238 of Stanley stating that Motzkin numbers equal to the ballot number without (1,-1,1). Since Ballot numbers are Catalan numbers, the result follows from the well-known bijection with Dyck paths.
a(n + 2) = p(n + 2) where p(x) is the unique degree-n polynomial such that p(k) = Catalan(k) for k = 1, 2, ..., n+1. - Michael Somos, Oct 07 2003
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; cf. p. 238.
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LINKS
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FORMULA
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G.f.: (sqrt( 1 - 2*x - 3*x^2 ) - sqrt( 1 - 4*x ) - x) / (2*x) = (4*x^2) / ( (1 - 2*x + sqrt( 1 - 4*x )) * (1 - x + sqrt( 1 - 2*x - 3*x^2 )) - 4*x^2). - Michael Somos, Jan 05 2012
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EXAMPLE
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x^2 + 3*x^3 + 10*x^4 + 33*x^5 + 111*x^6 + 378*x^7 + 1303*x^8 + 4539*x^9 + ...
a(4) = 10 since p(x) = x^2 - 2*x + 2 interpolates p(1) = 1, p(2) = 2, p(3) = 5, and p(4) = 10. - Michael Somos, Jan 05 2012
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PROG
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(PARI) {a(n) = if( n<2, 0, n--; subst( polinterpolate( vector(n, k, binomial( 2*k, k) / (k + 1))), x, n + 1))} /* Michael Somos, Jan 05 2012 */
(PARI) {a(n) = local(A); if( n<2, 0, n -= 2; A = x * O(x^n); polcoeff( 4 / ( (1 - 2*x + sqrt( 1 - 4*x + A )) * (1 - x + sqrt( 1 - 2*x - 3*x^2 + A)) - 4*x^2 ), n))} /* Michael Somos, Jan 05 2012 */
(PARI) { allocatemem(932245000); for (n = 1, 100, a=if(n<=1, 0, subst(polinterpolate(vector(n-1, k, binomial(2*k, k)/(k+1))), x, n)); write("b058987.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 24 2009
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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