

A000917


a(n) = (2n+3)!/(n!*(n+2)!).


8



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

0,1


COMMENTS

G.f.: c(x)*(4c(x))/(14*x)^(3/2), c(x) = g.f. for Catalan numbers A000108 (agrees with Han 75 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
Appears as diagonal in A003506.  Zerinvary Lajos, Apr 12 2006
a(n)=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


REFERENCES

E. R. Hansen, A Table of Series and Products, PrenticeHall, Englewood Cliffs, NJ, 1975, p. 99.


LINKS

T. D. Noe, Table of n, a(n) for n = 0..200


FORMULA

a(n) = (n+1)*binomial(2*n+3, n+1).  Vincenzo Librandi, Jun 01 2016


MAPLE

a := proc(n) (n+1)*binomial(2*n+3, n+2) end: seq(a(n), n=0..23); # Zerinvary Lajos, Nov 26 2006
seq((n+1)*binomial(2*n+4, n+2)/2, n=0..23); # Zerinvary Lajos, Feb 28 2007


MATHEMATICA

Table[(2*n + 3)!/(n!*(n + 2)!), {n, 0, 25}] (* T. D. Noe, Jun 20 2012 *)


PROG

(MAGMA) [(n+1)*Binomial(2*n+3, n+1): n in [0..25]]; // Vincenzo Librandi, Jun 01 2016


CROSSREFS

Cf. A007054, A038665, A038679, A000108, A000984, A000302, A003506. 1/beta(n, n+2) in A061928.
Sequence in context: A074831 A203357 A304494 * A025535 A119693 A158243
Adjacent sequences: A000914 A000915 A000916 * A000918 A000919 A000920


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


STATUS

approved



