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A379464 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -3), and staying on or above y = -2.

%I A379464 #20 Jan 29 2025 08:58:29
%S A379464 1,1,1,1,4,16,46,106,226,514,1306,3466,9002,22634,56330,142026,364743,
%T A379464 945303,2448511,6323695,16336885,42363693,110340297,288229377,
%U A379464 753920796,1973799396,5174280216,13588243696,35748326836,94188788164,248464963876,656148369796
%N A379464 a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -3), and staying on or above y = -2.
%H A379464 Robert Israel, <a href="/A379464/b379464.txt">Table of n, a(n) for n = 0..2271</a>
%F A379464 a(n) ~ 2^(5/2) * (1 + 4/3^(3/4))^(n + 3/2) / (3^(11/8) * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jan 15 2025
%F A379464 Conjecture D-finite with recurrence +3*(n+4)*(3*n+4)*(3*n+8)*a(n) +3*(-63*n^3-297*n^2-349*n-60)*a(n-1) +3*(189*n^3+270*n^2-229*n-140)*a(n-2) +15*(-63*n^3+117*n^2+44*n-64)*a(n-3) +(689*n^3-5372*n^2+6946*n-1288)*a(n-4) +(n-4)*(201*n^2+2767*n-3011)*a(n-5) -(n-5)*(579*n+257)*(n-4)*a(n-6) +229*(n-5)*(n-6)*(n-4)*a(n-7)=0. - _R. J. Mathar_, Jan 29 2025
%e A379464 For n = 4, the a(4)=4 paths are HHHH, UDUU, UUDU, UUUD, where U=(1,1), D=(1,-3) and H=(1,0).
%p A379464 f:= proc(n,y) option remember;
%p A379464     if n = 0 then if y = 0 then return 1 else return 0 fi fi;
%p A379464     if y > n then return 0 fi;
%p A379464     if y >= -1 then procname(n-1,y-1) + procname(n-1,y) + procname(n-1,y+3)
%p A379464     else procname(n-1,y) + procname(n-1,y+3)
%p A379464     fi;
%p A379464 end proc:
%p A379464 map(f, [$0..40],0); # _Robert Israel_, Jan 23 2025
%o A379464 (PARI) a(n) = sum(k=0, floor(n/4), 3*binomial(n, k*4)*binomial(4*k+3, k)/(4*k+3)) \\ _Thomas Scheuerle_, Jan 07 2025
%Y A379464 Cf. A127902, A379463.
%K A379464 nonn
%O A379464 0,5
%A A379464 _Emely Hanna Li Lobnig_, Dec 23 2024
%E A379464 More terms from _Jinyuan Wang_, Jan 07 2025