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A134424 Area under all paths in the first quadrant from (0,0) to (n,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0).

%I A134424 #12 Sep 25 2024 01:53:30
%S A134424 0,0,1,4,21,80,316,1152,4186,14812,52020,180616,623338,2138040,
%T A134424 7302035,24842736,84262609,285052676,962184359,3241616628,10903119167,
%U A134424 36619715860,122837641530,411588875136,1377735161776,4607695277512
%N A134424 Area under all paths in the first quadrant from (0,0) to (n,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0).
%F A134424 a(n) = Sum_{k>=0} k * A134423(n,k).
%F A134424 G.f.: z^2*(1+z^2)*g^2/((1+z-z^2)*(1-3*z-z^2)), where g=1+z*g+z^2*g+z^2*g^2 (g is the g.f. of A128720).
%F A134424 Conjecture D-finite with recurrence -(n+2)*(5*n-7)*a(n) -(n+1)*(5*n-127)*a(n-1) +(135*n^2-655*n-42)*a(n-2) +2*(5*n^2-275*n-108)*a(n-3) +(-725*n^2+4941*n-5734)*a(n-4) +(-235*n^2+1880*n-1173)*a(n-5) +(725*n^2-6659*n+12606)*a(n-6) +2*(5*n^2+195*n-1988)*a(n-7) +(-135*n^2+1505*n-3358)*a(n-8) -(5*n+87)*(n-9)*a(n-9) +(5*n-33)*(n-10)*a(n-10)=0. - _R. J. Mathar_, Jul 24 2022
%e A134424 a(3)=4 because the areas under the paths hhh, hH, Hh, hUD, UhD and UDh are 0,0,0,1,2 and 1, respectively.
%p A134424 g:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/z^2: G:=z^2*(1+z^2)*g^2/((1+z-z^2)*(1-3*z-z^2)): Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..25);
%Y A134424 Cf. A128720, A134423.
%K A134424 nonn
%O A134424 0,4
%A A134424 _Emeric Deutsch_, Oct 25 2007