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A257390 Number of 4-Motzkin paths of length n with no level steps at even level.

%I A257390 #16 Apr 09 2017 03:40:33
%S A257390 1,0,1,4,18,80,357,1596,7150,32096,144362,650568,2937316,13286368,
%T A257390 60205805,273290988,1242639446,5659468736,25816338046,117945079736,
%U A257390 539646216188,2472638868960,11345220210658,52124831171544,239792244636876,1104495824173376
%N A257390 Number of 4-Motzkin paths of length n with no level steps at even level.
%H A257390 Robert Israel, <a href="/A257390/b257390.txt">Table of n, a(n) for n = 0..1322</a>
%F A257390 a(n) = Sum_{i=0..floor(n/2)}4^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the i-th Catalan number A000108.
%F A257390 G.f.: (1-4*x-sqrt((1-4*x)*(1-4*x-4*x^2)))/(2*x^2).
%F A257390 a(n) ~ 2^(n+3/4) * (1+sqrt(2))^(n+1/2) / (sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Apr 22 2015
%F A257390 a(n) = ((-16*n + 40)*a(n-3) + (-12*n+12)*a(n-2) +(8*n+4)*a(n-1))/(n+2). - _Robert Israel_, Apr 22 2015
%p A257390 rec:= a(n) = ((-16*n + 40)*a(n-3) + (-12*n+12)*a(n-2) +(8*n+4)*a(n-1))/(n+2):
%p A257390 f:= gfun:-rectoproc({rec,a(0)=1,a(1)=0,a(2)=1},a(n),remember):
%p A257390 seq(f(i),i=0..100); # _Robert Israel_, Apr 22 2015
%t A257390 CoefficientList[Series[(1-4*x-Sqrt[(1-4*x)*(1-4*x-4*x^2)])/(2*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 22 2015 *)
%o A257390 (PARI) x='x+O('x^50); Vec((1-4*x-sqrt((1-4*x)*(1-4*x-4*x^2)))/(2*x^2)) \\ _G. C. Greubel_, Apr 08 2017
%Y A257390 Cf. A090345, A025266, A257290.
%K A257390 nonn
%O A257390 0,4
%A A257390 _José Luis Ramírez Ramírez_, Apr 21 2015