A030060 Third derivative of Catalan generating function/3!.

5, 56, 420, 2640, 15015, 80080, 408408, 2015520, 9699690, 45762640, 212469400, 973496160, 4411154475, 19800295200, 88158457200, 389753179200, 1712478031110, 7483097278800, 32540135136600, 140883148005600, 607558575774150, 2610765994183776, 11182476723339600

Offset

0,1

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Formula

a(n) = (2*(n+3))!/(3!*n!*(n+4)!) = (n+1)*(n+2)*(n+3)*C(n+3)/6, C(n): Catalan numbers.

From Amiram Eldar, Mar 22 2022: (Start)

Sum_{n>=0} 1/a(n) = (sqrt(3)*Pi - 5)/2.

Sum_{n>=0} (-1)^n/a(n) = 9*sqrt(5)*log(phi) - 19/2, where phi is the golden ratio (A001622). (End)

Mathematica

Array[CatalanNumber[#] Binomial[#, 3] &, 19, 3] (* Michael De Vlieger, Dec 17 2017 *)

Prog

(MuPAD) combinat::catalan(n) *binomial(n, 3) $ n = 3..21 // Zerinvary Lajos, Feb 15 2007
(PARI) a(n) = (2*(n+3))!/(3!*n!*(n+4)!) \\ Andrew Howroyd, Dec 17 2017

Crossrefs

Cf. A000108, A001622.

Sequence in context: A041995 A288543 A062125 * A247710 A247774 A258490

Adjacent sequences: A030057 A030058 A030059 * A030061 A030062 A030063

Keyword

nonn

Author

Wolfdieter Lang

Status

approved