A134718 Even Motzkin numbers.

2, 4, 2188, 5798, 113634, 310572, 6536382, 18199284, 25669818476, 73007772802, 114988706524270, 330931069469828, 556704809728838604, 1614282136160911722, 39532221379621112004, 114956499435014161638, 2837208756709314025578, 8270140811590103129028, 14996791899280244858336604

Offset: 1

Author

Omar E. Pol, Nov 11 2007

Keywords

• easy
• nonn

Comments

The values of n such that the Motzkin number M(n) (=A001006(n)) is even are given in A081706. - Emeric Deutsch, Dec 07 2007

A001006 except A134717. - Vladimir Reshetnikov, Nov 02 2015

The asymptotic density of this sequence within the Motzkin numbers is 1/3. - Amiram Eldar, Aug 26 2024

Links

• G. C. Greubel, Table of n, a(n) for n = 1..300
• Emeric Deutsch and Bruce E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory, Vol. 117, No. 1 (2006), 191-215.

Crossrefs

Cf. A001006, A081706, A134717.

Programs

Maple

M := n -> add(binomial(n, 2*k)*binomial(2*k,k)/(k+1), k=0..n):
a := n -> `if`(`mod`(M(n),2)=0, M(n), NULL);
seq(a(n), n=0..50); # Emeric Deutsch, Dec 07 2007

Mathematica

Select[Table[(-1)^n Hypergeometric2F1[3/2, -n, 3, 4], {n, 0, 60}], EvenQ] (* Vladimir Reshetnikov, Nov 02 2015 *)

PARI

a001006(n) = polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/ (2*x^2), n); for(n=0, 100, if((m=a001006(n))%2==0, print1(m", "))) \\ Altug Alkan, Nov 03 2015

Formula

a(n) = A001006(A081706(n)). - Amiram Eldar, Aug 26 2024

Extensions

More terms from Emeric Deutsch, Dec 07 2007

a(91) in b-file corrected by Andrew Howroyd, Feb 23 2018