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A134718 Even Motzkin numbers.

%I A134718 #27 Aug 26 2024 04:58:52
%S A134718 2,4,2188,5798,113634,310572,6536382,18199284,25669818476,73007772802,
%T A134718 114988706524270,330931069469828,556704809728838604,
%U A134718 1614282136160911722,39532221379621112004,114956499435014161638,2837208756709314025578,8270140811590103129028,14996791899280244858336604
%N A134718 Even Motzkin numbers.
%C A134718 The values of n such that the Motzkin number M(n) (=A001006(n)) is even are given in A081706. - _Emeric Deutsch_, Dec 07 2007
%C A134718 A001006 except A134717. - _Vladimir Reshetnikov_, Nov 02 2015
%C A134718 The asymptotic density of this sequence within the Motzkin numbers is 1/3. - _Amiram Eldar_, Aug 26 2024
%H A134718 G. C. Greubel, <a href="/A134718/b134718.txt">Table of n, a(n) for n = 1..300</a>
%H A134718 Emeric Deutsch and Bruce E. Sagan, <a href="http://dx.doi.org/10.1016/j.jnt.2005.06.005">Congruences for Catalan and Motzkin numbers and related sequences</a>, J. Num. Theory, Vol. 117, No. 1 (2006), 191-215.
%F A134718 a(n) = A001006(A081706(n)). - _Amiram Eldar_, Aug 26 2024
%p A134718 M := n -> add(binomial(n, 2*k)*binomial(2*k,k)/(k+1), k=0..n):
%p A134718 a := n -> `if`(`mod`(M(n),2)=0, M(n), NULL);
%p A134718 seq(a(n), n=0..50); # _Emeric Deutsch_, Dec 07 2007
%t A134718 Select[Table[(-1)^n Hypergeometric2F1[3/2, -n, 3, 4], {n, 0, 60}], EvenQ] (* _Vladimir Reshetnikov_, Nov 02 2015 *)
%o A134718 (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
%Y A134718 Cf. A001006, A081706, A134717.
%K A134718 easy,nonn
%O A134718 1,1
%A A134718 _Omar E. Pol_, Nov 11 2007
%E A134718 More terms from _Emeric Deutsch_, Dec 07 2007
%E A134718 a(91) in b-file corrected by _Andrew Howroyd_, Feb 23 2018