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A379463 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 = -1.

%I A379463 #16 Jan 29 2025 08:53:06
%S A379463 1,1,1,1,3,11,31,71,150,334,826,2146,5498,13690,33762,84306,214451,
%T A379463 551107,1417291,3637627,9343555,24096675,62439587,162331747,422773098,
%U A379463 1102422546,2879207046,7534606366,19756893196,51894005428,136496647696,359478351816,947912008073
%N A379463 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 = -1.
%F A379463 a(n) ~ 2^(3/2) * (1 + 4/3^(3/4))^(n + 3/2) / (3^(11/8) * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jan 15 2025
%F A379463 Conjecture D-finite with recurrence 3*n*(3*n+4)*(n-3)*(3*n+8)*a(n) +3*(-45*n^4+54*n^3+192*n^2-27*n-20)*a(n-1)
%F A379463 +9*(n-1)*(30*n^3-72*n^2-7*n+20)*a(n-2) -3*(n-1)*(n-2)*(90*n^2-234*n+95)*a(n-3) -(n-1)*(n-2)*(n-3)*(121*n+499)*a(n-4) +229*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Jan 29 2025
%e A379463 For n = 4, the a(4)=3 paths are HHHH, UUDU, UUUD, where U=(1,1), D=(1,-3) and H=(1,0).
%p A379463 A379463 := proc(n)
%p A379463     add(2*binomial(n, k*4)*binomial(4*k+1, k)/(3*k+2),k=0..floor(n/4)) ;
%p A379463 end proc:
%p A379463 seq(A379463(n),n=0..50) ; # _R. J. Mathar_, Jan 29 2025
%o A379463 (PARI) a(n) = sum(k=0, floor(n/4), 2*binomial(n, k*4)*binomial(4*k+1, k)/(3*k+2)) \\ _Thomas Scheuerle_, Jan 07 2025
%Y A379463 Cf. A069271, A127902, A379464.
%K A379463 nonn
%O A379463 0,5
%A A379463 _Emely Hanna Li Lobnig_, Dec 23 2024
%E A379463 More terms from _Jinyuan Wang_, Jan 07 2025