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A000917
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a(n) = (2n+3)!/(n!*(n+2)!).
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9
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3, 20, 105, 504, 2310, 10296, 45045, 194480, 831402, 3527160, 14872858, 62403600, 260757900, 1085822640, 4508102925, 18668849760, 77138650050, 318107374200, 1309542023790, 5382578744400, 22093039119060, 90567738003600, 370847442355650, 1516927277253024
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OFFSET
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0,1
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COMMENTS
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G.f.: c(x)*(4-c(x))/(1-4*x)^(3/2), c(x) = g.f. for Catalan numbers A000108 (agrees with Hansen, 1975, p. 99, (5.27.9)). Convolution of A038679 with A000984 (central binomial coefficients); also convolution of A038665 with A000302 (powers of 4). - Wolfdieter Lang, Dec 11 1999
a(n) is the number of double rises in all Grand Dyck paths of semilength n+2. Example: a(0)=3 because in the 6 (=A000984(2)) Grand Dyck paths of semilength 2, namely udud, (uu)dd, uddu, d(uu)d, dudu, dd(uu), we have a total of 3 uu's (shown between parentheses). - Emeric Deutsch, Nov 29 2008
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REFERENCES
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Eldon R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 99, (5.27.9).
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LINKS
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FORMULA
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D-finite with recurrence +(n+2)*a(n) +10*(-n-1)*a(n-1) +12*(2*n+1)*a(n-2)=0. - R. J. Mathar, Nov 09 2021
D-finite with recurrence n*(n+2)*a(n) -2*(2*n+3)*(n+1)*a(n-1)=0. - R. J. Mathar, Nov 09 2021
Sum_{n>=0} 1/a(n) = 1 - Pi/(3*sqrt(3)) = 1 - A073010.
Sum_{n>=0} (-1)^n/a(n) = 6*log(phi)/sqrt(5) - 1, where phi is the golden ratio (A001622). (End)
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MAPLE
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a := proc(n) (n+1)*binomial(2*n+3, n+2) end: seq(a(n), n=0..23); # Zerinvary Lajos, Nov 26 2006
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MATHEMATICA
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Table[(2*n + 3)!/(n!*(n + 2)!), {n, 0, 25}] (* T. D. Noe, Jun 20 2012 *)
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PROG
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(Magma) [(n+1)*Binomial(2*n+3, n+1): n in [0..25]]; // Vincenzo Librandi, Jun 01 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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