%I #53 Sep 08 2022 08:44:28
%S 3,20,105,504,2310,10296,45045,194480,831402,3527160,14872858,
%T 62403600,260757900,1085822640,4508102925,18668849760,77138650050,
%U 318107374200,1309542023790,5382578744400,22093039119060,90567738003600,370847442355650,1516927277253024
%N a(n) = (2n+3)!/(n!*(n+2)!).
%C 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
%C Appears as diagonal in A003506. - _Zerinvary Lajos_, Apr 12 2006
%C 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
%D Eldon R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, NJ, 1975, p. 99, (5.27.9).
%H T. D. Noe, <a href="/A000917/b000917.txt">Table of n, a(n) for n = 0..200</a>
%H Jian Zhou, <a href="https://arxiv.org/abs/2108.10514">On Some Mathematics Related to the Interpolating Statistics</a>, arXiv:2108.10514 [math-ph], 2021.
%F a(n) = (n+1)*binomial(2*n+3, n+1) = (n+1)*A001700(n+1). - _Vincenzo Librandi_, Jun 01 2016
%F a(n) = (2*n+3)*A001791(n+1). - _R. J. Mathar_, Nov 09 2021
%F 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
%F 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
%F From _Amiram Eldar_, Jan 24 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 1 - Pi/(3*sqrt(3)) = 1 - A073010.
%F Sum_{n>=0} (-1)^n/a(n) = 6*log(phi)/sqrt(5) - 1, where phi is the golden ratio (A001622). (End)
%p a := proc(n) (n+1)*binomial(2*n+3, n+2) end: seq(a(n), n=0..23); # _Zerinvary Lajos_, Nov 26 2006
%p seq((n+1)*binomial(2*n+4, n+2)/2, n=0..23); # _Zerinvary Lajos_, Feb 28 2007
%t Table[(2*n + 3)!/(n!*(n + 2)!), {n, 0, 25}] (* _T. D. Noe_, Jun 20 2012 *)
%o (Magma) [(n+1)*Binomial(2*n+3, n+1): n in [0..25]]; // _Vincenzo Librandi_, Jun 01 2016
%Y Cf. A001622, A001791, A007054, A038665, A038679, A000108, A000984, A000302, A003506, A073010.
%Y 1/beta(n, n+2) in A061928.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_