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%I #19 Jul 14 2021 14:50:50
%S 0,2,2,6,12,30,72,182,464,1206,3170,8426,22596,61074,166194,454950,
%T 1251984,3461574,9611190,26787378,74916660,210178458,591347988,
%U 1668172842,4717282752,13369522250,37970114702,108045430902
%N a(n) = 3*trinomial(n+1,0) - trinomial(n+2,0).
%C First differs from A059727 for n = 8.
%C Essentially twice A005043.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>
%p trinomial := n -> simplify(GegenbauerC(n,-n,-1/2)):
%p a := n -> 3*trinomial(n+1) - trinomial(n+2):
%p seq(a(n), n=0..27); # _Peter Luschny_, May 07 2016
%t Table[(4*2^n (2n + 3)!! (3 Hypergeometric2F1[-2 - n, -1 - n, -3/2 - n, 1/4] - 4 Hypergeometric2F1[-2 - n, -2 - n, -3/2 - n, 1/4]))/(n + 2)!, {n, 0, 20}] (* _Vladimir Reshetnikov_, May 07 2016 *)
%Y Cf. A005043, A059727.
%K nonn,easy
%O 0,2
%A _Eric W. Weisstein_, Feb 19 2005
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