A224791 Triangle T(n,k) read by rows: left edge is 0, 1, 2, ... (cf. A001477); otherwise each entry is sum of entry to left and entries immediately above it to left and right, with 1 for the missing right term at right edge.

0, 1, 2, 2, 5, 8, 3, 10, 23, 32, 4, 17, 50, 105, 138, 5, 26, 93, 248, 491, 630, 6, 37, 156, 497, 1236, 2357, 2988, 7, 50, 243, 896, 2629, 6222, 11567, 14556, 8, 65, 358, 1497, 5022, 13873, 31662, 57785, 72342, 9, 82, 505, 2360, 8879, 27774, 73309, 162756

Offset: 0

Author

Reinhard Zumkeller, Apr 18 2013

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  0;
  1,  2;
  2,  5,  8;
  3, 10, 23,  32;
  4, 17, 50, 105, 138;
		

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A051601, A059283.

Programs

Haskell

a224791 n k = a224791_tabl !! n !! k
a224791_row n = a224791_tabl !! n
a224791_tabl = iterate
   (\row -> scanl1 (+) $ zipWith (+) ([1] ++ row) (row ++ [1])) [0]

Maple

T:= proc(n, k) option remember;
      if k=0 then n
    elif k=n then T(n,n-1) + T(n-1,n-1) + 1
    else T(n,k-1) + T(n-1,k-1) + T(n-1, k)
      fi
    end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 12 2019

Mathematica

T[n_, k_]:= T[n, k]= If[k==0, n, If[k==n , T[n, n-1] + T[n-1, n-1] + 1, T[n, k-1] + T[n-1, k-1] + T[n-1, k]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 12 2019 *)

PARI

T(n,k) = if(k==0, n, if(k==n, T(n,n-1) + T(n-1,n-1) + 1, T(n,k-1) + T(n-1,k-1) + T(n-1, k) )); \\ G. C. Greubel, Nov 12 2019

Sage

@CachedFunction
def T(n, k):
    if (k==0): return n
    elif (k==n): return T(n,n-1) + T(n-1,n-1) + 1
    else: return T(n,k-1) + T(n-1,k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 12 2019

Formula

T(n,0) = n, T(n+1,k) = T(n+1,k-1) + T(n,k-1) + T(n,k) (0 < k <= n) and T(n+1,n+1) = T(n+1,n) + T(n,n) + 1.