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A055450 Path-counting array T; each step of a path is (1 right) or (1 up) to a point below line y=x, else (1 right and 1 up) or (1 up) to a point on the line y=x, else (1 left) or (1 up) to a point above line y=x. T(i,j)=number of paths to point (i-j,j), for 1<=j<=i, i >= 1.

%I A055450 #20 Jan 31 2024 16:56:50
%S A055450 1,1,3,1,2,10,1,3,7,36,1,4,5,26,137,1,5,9,19,101,543,1,6,14,14,75,406,
%T A055450 2219,1,7,20,28,56,305,1676,9285,1,8,27,48,42,230,1270,7066,39587,1,9,
%U A055450 35,75,90,174,965,5390,30302,171369,1,10,44,110,165,132,735,4120,23236,131782,751236
%N A055450 Path-counting array T; each step of a path is (1 right) or (1 up) to a point below line y=x, else (1 right and 1 up) or (1 up) to a point on the line y=x, else (1 left) or (1 up) to a point above line y=x. T(i,j)=number of paths to point (i-j,j), for 1<=j<=i, i >= 1.
%H A055450 G. C. Greubel, <a href="/A055450/b055450.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A055450 Initial values: T(i, 0)=1 for i >= 0. Recurrence: if 1 <= j < i/2, then T(i, j) = T(i-1, j-1) + T(i-1, j), if j = i/2 then T(2j, j) = T(2j-2, j-1) + T(2j-1, j-1), otherwise T(2j-k, j) = T(2j-k+1, j) + T(2j-k-1, j-1) for j=k, k+1, k+2, ..., for k=1, 2, 3, ...
%F A055450 T(2n, n) = A000108(n) for n >= 0 (Catalan numbers).
%F A055450 T(n, n) = A002212(n+1).
%F A055450 T(n, n-1) = A045868(n).
%F A055450 T(n, k) = A030237(n-k+1, k) for n >= 1, 0 <= k < n/2.
%F A055450 From _G. C. Greubel_, Jan 29 2024: (Start)
%F A055450 T(n, k) = (n-2*k+2)*binomial(n+1, k)/(n-k+2) for 0 <= k < n/2, otherwise Catalan(n-k +1)*Hypergeometric2F1([n-2*k, (3+2*(n-k))/2], [3+n-k], -4).
%F A055450 Sum_{k=0..n} T(n, k) = A055451(n). (End)
%e A055450 Triangle begins as:
%e A055450   1;
%e A055450   1, 3;
%e A055450   1, 2, 10;
%e A055450   1, 3,  7, 36;
%e A055450   1, 4,  5, 26, 137;
%e A055450   1, 5,  9, 19, 101, 543;
%e A055450   1, 6, 14, 14,  75, 406, 2219;
%e A055450   1, 7, 20, 28,  56, 305, 1676, 9285;
%e A055450   1, 8, 27, 48,  42, 230, 1270, 7066, 39587;
%e A055450   ...
%e A055450 T(4,4) defined as T(5,4)+T(3,3) when k=4, T(5,4) already defined when k=3.
%t A055450 T[n_, 0]:= 1; T[n_, k_]:= T[n, k]= If[1<=k<n/2, T[n-1, k-1] + T[n-1, k], If[k==n/2, T[n-2, k-1] + T[n-1, k-1], T[n+1, k] + T[n-1, k-1] ]];
%t A055450 Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 29 2024 *)
%t A055450 T[n_, k_]:= If[k<n/2, (n-2*k+2)*Binomial[n+1,k]/(n-k+2), CatalanNumber[ n-k+1]*Hypergeometric2F1[n-2*k, (3+2*(n-k))/2, 3+n-k,-4]];
%t A055450 Table[T[n,k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 29 2024 *)
%o A055450 (Magma)
%o A055450 B:=Binomial; G:=Gamma; F:=Factorial;
%o A055450 p:= func< n,k,j | B(n-2*k+j-1, j)*G(n-k+j+3/2)/(F(j)*G(n-k+3/2)*B(n-k+j+2, j)) >;
%o A055450 A030237:= func< n,k | (n-k+1)*Binomial(n+k, k)/(n+1) >;
%o A055450 function T(n,k) // T = A055450
%o A055450   if k lt n/2 then return A030237(n-k+1, k);
%o A055450   else return Round(Catalan(n-k+1)*(&+[p(n,k,j)*(-4)^j: j in [0..n]]));
%o A055450   end if;
%o A055450 end function;
%o A055450 [T(n,k): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Jan 29 2024
%o A055450 (SageMath)
%o A055450 def A030237(n,k): return (n-k+1)*binomial(n+k, k)/(n+1)
%o A055450 def T(n,k): # T = A055450
%o A055450     if k<n/2: return A030237(n-k+1,k)
%o A055450     else: return round(catalan_number(n-k+1)*hypergeometric([n-2*k, (3+2*(n-k))/2], [3+n-k], -4))
%o A055450 flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jan 29 2024
%Y A055450 Cf. A000108, A002212, A030237, A045868,
%Y A055450 Cf. A055451, A055452, A055453, A055454, A055455.
%K A055450 nonn,tabl
%O A055450 0,3
%A A055450 _Clark Kimberling_, May 18 2000