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%I #24 Sep 08 2022 08:45:29
%S 1,4,14,48,162,544,1820,6080,20290,67680,225684,752448,2508468,
%T 8362176,27875064,92919168,309734850,1032458080,3441543140,
%U 11471842880,38239537852,127465249344,424884399624,1416281802368,4720940242612,15736469278144,52454901060680
%N a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*3^(n-k).
%C Hankel transform is (-2)^n. In general, given r>=0, the sequence given by Sum_{k=0..n} C(n,floor(k/2))*r^(n-k) has Hankel transform (1-r)^n. The sequence is the image of the sequence with g.f. (1+x)/(1-3x) under the Chebyshev mapping g(x)->(1/sqrt(1-4x^2))*g(xc(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.
%H Vincenzo Librandi, <a href="/A127359/b127359.txt">Table of n, a(n) for n = 0..300</a>
%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%F G.f.: (1/sqrt(1-4x^2))*(1+x*c(x^2))/(1-3*x*c(x^2)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x)).
%F a(n) = Sum_{k=0..n} A061554(n,k)*3^k. - _Philippe Deléham_, Dec 04 2009
%F Recurrence: 3*n*a(n) = 2*(5*n + 3)*a(n-1) + 4*(3*n - 11)*a(n-2) - 40*(n-2)*a(n-3). - _Vaclav Kotesovec_, Oct 19 2012
%F a(n) ~ 4*10^n/3^(n+1). - _Vaclav Kotesovec_, Oct 19 2012
%p A127359:=n->sum(binomial(n,floor(k/2))*3^(n-k), k=0..n): seq(A127359(n), n=0..30); # _Wesley Ivan Hurt_, Mar 14 2015
%t Table[Sum[Binomial[n, Floor[k/2]]*3^(n-k), {k,0,n}], {n,0,30}] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%o (PARI) a(n) = sum(j=0, n, binomial(n, j\2)*3^(n-j));
%o vector(31, n, a(n-1)) \\ _G. C. Greubel_, Dec 15 2019
%o (Magma) I:=[1, 4, 14]; [n le 3 select I[n] else (2*(5*n-2)*Self(n-1) + 4*(3*n - 14)*Self(n-2) -40*(n-3)*Self(n-3))/(3*(n-1)): n in [1..30]]; // _G. C. Greubel_, Dec 15 2019
%o (Sage) [sum(binomial(n, floor(j/2))*3^(n-j) for j in (0..n)) for n in (0..30)] # _G. C. Greubel_, Dec 15 2019
%o (GAP) a:=[1, 4, 14];; for n in [4..30] do a[n]:=(2*(5*n-2)*a[n-1] +4*(3*n-14)*a[n-2] -40*(n-3)*a[n-3])/(3*(n-1)); od; a; # _G. C. Greubel_, Dec 15 2019
%Y Cf. A107430. - _Philippe Deléham_, Sep 16 2009
%Y Cf. A000108 (Catalan numbers).
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jan 11 2007
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