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.

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, 2219, 1, 7, 20, 28, 56, 305, 1676, 9285, 1, 8, 27, 48, 42, 230, 1270, 7066, 39587, 1, 9, 35, 75, 90, 174, 965, 5390, 30302, 171369, 1, 10, 44, 110, 165, 132, 735, 4120, 23236, 131782, 751236

Offset

0,3

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

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, ...

T(2n, n) = A000108(n) for n >= 0 (Catalan numbers).

T(n, n) = A002212(n+1).

T(n, n-1) = A045868(n).

T(n, k) = A030237(n-k+1, k) for n >= 1, 0 <= k < n/2.

From G. C. Greubel, Jan 29 2024: (Start)

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).

Sum_{k=0..n} T(n, k) = A055451(n). (End)

Example

Triangle begins as:
  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, 2219;
  1, 7, 20, 28,  56, 305, 1676, 9285;
  1, 8, 27, 48,  42, 230, 1270, 7066, 39587;
  ...
T(4,4) defined as T(5,4)+T(3,3) when k=4, T(5,4) already defined when k=3.

Mathematica

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] ]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 29 2024 *)
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]];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 29 2024 *)

Prog

(Magma)
B:=Binomial; G:=Gamma; F:=Factorial;
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)) >;
A030237:= func< n, k | (n-k+1)*Binomial(n+k, k)/(n+1) >;
function T(n, k) // T = A055450
  if k lt n/2 then return A030237(n-k+1, k);
  else return Round(Catalan(n-k+1)*(&+[p(n, k, j)*(-4)^j: j in [0..n]]));
  end if;
end function;
[T(n, k): k in [0..n], n in [0..13]]; // G. C. Greubel, Jan 29 2024
(SageMath)
def A030237(n, k): return (n-k+1)*binomial(n+k, k)/(n+1)
def T(n, k): # T = A055450
    if k<n/2: return A030237(n-k+1, k)
    else: return round(catalan_number(n-k+1)*hypergeometric([n-2*k, (3+2*(n-k))/2], [3+n-k], -4))
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 29 2024

Crossrefs

Cf. A000108, A002212, A030237, A045868,

Cf. A055451, A055452, A055453, A055454, A055455.

Sequence in context: A132323 A305097 A316757 * A185835 A301669 A300427

Adjacent sequences: A055447 A055448 A055449 * A055451 A055452 A055453

Keyword

nonn,tabl

Author

Clark Kimberling, May 18 2000

Status

approved