A000917 a(n) = (2n+3)!/(n!*(n+2)!).

3, 20, 105, 504, 2310, 10296, 45045, 194480, 831402, 3527160, 14872858, 62403600, 260757900, 1085822640, 4508102925, 18668849760, 77138650050, 318107374200, 1309542023790, 5382578744400, 22093039119060, 90567738003600, 370847442355650, 1516927277253024

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Appears as diagonal in A003506. - Zerinvary Lajos, Apr 12 2006

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

References

• Eldon R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 99, (5.27.9).

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Crossrefs

Cf. A001622, A001791, A007054, A038665, A038679, A000108, A000984, A000302, A003506, A073010.

1/beta(n, n+2) in A061928.

Programs

Magma

[(n+1)*Binomial(2*n+3, n+1): n in [0..25]]; // 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 *)

Formula

a(n) = (n+1)*binomial(2*n+3, n+1) = (n+1)*A001700(n+1). - Vincenzo Librandi, Jun 01 2016

a(n) = (2*n+3)*A001791(n+1). - R. J. Mathar, Nov 09 2021

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

From Amiram Eldar, Jan 24 2022: (Start)

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)