A259550 a(n) = C(5*n-1,2*n)/3, n > 0, a(0) = 1.

1, 2, 42, 1001, 25194, 653752, 17298645, 463991880, 12570420330, 343176898988, 9425842448792, 260170725132045, 7210477496434485, 200519284375732896, 5592628786362932776, 156375886125188595376, 4382048530314336892010, 123033460966787345446836

Offset

0,2

Links

Table of n, a(n) for n=0..17.

D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.

V. V. Kruchinin and D. V. Kruchinin, Composita and its properties, J. Analysis and Number Theory 2 (2014), 1-8.

V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6.

D. V. Kruchinin, On solving some functional equations, Advances in Difference Equations, Vol. 1 (2015), 1687-1847.

Formula

G.f.: A(x) = 1 + (x*B(x)')/(B(x)), B(x) = (1 + x*B(x)^5)*(C(x*B(x)^5), C(x) is g.f. of Catalan numbers.

a(n) = n*Sum_{i = 0..n}((C(5*n,i)*C(7*n-2*i-1,n-i))/(6*n-i)), n > 1, a(0) = 1.

a(n) = 1/5*A001450(n) for n >= 1. exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*x + 23*x^2 + 377*x^3 + ... is the o.g.f. for the sequence of Duchon numbers A060941. - Peter Bala, Oct 05 2015

Mathematica

Join[{1}, Table[Binomial[5 n - 1, 2 n]/3, {n, 30}]] (* Vincenzo Librandi, Jul 01 2015 *)

Prog

(Maxima)
makelist(if n=0 then 1 else binomial(5*n-1, 2*n)/3, n, 0, 20);
(PARI) vector(20, n, n--; if (n==0, 1, binomial(5*n-1, 2*n)/3)) \\ Michel Marcus, Jul 01 2015
(Magma) [1] cat [Binomial(5*n-1, 2*n)/3: n in [1..20]]; // Vincenzo Librandi, Jul 01 2015

Crossrefs

Cf. A000108, A167422, A001450, A060941.

Sequence in context: A308526 A162678 A265867 * A177456 A360238 A216029

Adjacent sequences: A259547 A259548 A259549 * A259551 A259552 A259553

Keyword

nonn,easy

Author

Vladimir Kruchinin, Jun 30 2015

Extensions

More terms from Vincenzo Librandi, Jul 01 2015

Status

approved