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A000917
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(2n+3)!/(n!*(n+2)!).
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4
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Comment from wolfdieter.lang(AT)physik.uni-karlsruhe.de: G.f.: c(x)*(4-c(x))/(1-4*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).
Appears as diagonal in A003506. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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). [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 29 2008]
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REFERENCES
| E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 99.
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MAPLE
| a:=proc(n) if n=1 then 3 else binomial(2*n+1, n+1)*n fi end: seq(a(n), n=0..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 26 2006
seq((n-1)*binomial(2*n, n)/2, n=2..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 28 2007
a:=n->add(binomial(2*n, n)/2, k=2..n): seq(a(n), n=2..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007
a:=n->abs(sum((binomial(-n, n-4)), j=2..n)): seq(a(n), n=4..24); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 03 2007
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CROSSREFS
| Cf. A007054, A038665, A038679, A000108, A000984, A000302. 1/beta(n, n+2) in A061928.
Cf. A003506.
Cf. A000984.
Sequence in context: A165960 A074831 A203357 * A025535 A119693 A158243
Adjacent sequences: A000914 A000915 A000916 * A000918 A000919 A000920
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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