A055818 Triangle T read by rows: T(i,j) = R(i-j,j), where R(i,0) = R(0,i) = 1 for i >= 0, R(i,j) = Sum_{h=0..i-1} Sum_{m=0..j} R(h,m) for i >= 1, j >= 1.

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 11, 9, 4, 1, 1, 23, 24, 14, 5, 1, 1, 47, 60, 43, 20, 6, 1, 1, 95, 144, 122, 69, 27, 7, 1, 1, 191, 336, 328, 217, 103, 35, 8, 1, 1, 383, 768, 848, 640, 354, 146, 44, 9, 1, 1, 767, 1728, 2128, 1800, 1131, 543, 199, 54, 10, 1

Offset: 0

Author

Clark Kimberling, May 28 2000

Keywords

• nonn
• tabl

Examples

			Rows begins as:
  1;
  1,  1;
  1,  2, 1;
  1,  5, 3, 1;
  1, 11, 9, 4, 1;
  ...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 3B.

Crossrefs

Cf. A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055826, A055827, A055828, A055829.

Programs

GAP

T:= function(i,j)
    if i=0 or j=0 then return 1;
    else return Sum([0..i-1], h-> Sum([0..j], m-> T(h,m) ));
    fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n-k,k) ))); # G. C. Greubel, Jan 21 2020

Magma

function T(i,j)
  if i eq 0 or j eq 0 then return 1;
  else return (&+[(&+[T(h,m): m in [0..j]]): h in [0..i-1]]);
  end if; return T; end function;
[T(n-k,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 21 2020

Maple

T:= proc(i, j) option remember;
      if i=0 or j=0 then 1
    else add(add(T(h,m), m=0..j), h=0..i-1)
      fi; end:
seq(seq(T(n-k, k), k=0..n), n=0..12); # G. C. Greubel, Jan 21 2020

Mathematica

T[i_, j_]:= T[i, j]= If[i==0 || j==0, 1, Sum[T[h, m], {h,0,i-1}, {m,0,j}]]; Table[T[n-k, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 21 2020 *)

PARI

T(i,j) = if(i==0 || j==0, 1, sum(h=0,i-1, sum(m=0,j, T(h,m) )));
for(n=0,12, for(k=0, n, print1(T(n-k,k), ", "))) \\ G. C. Greubel, Jan 21 2020

Sage

@CachedFunction
def T(i, j):
    if (i==0 or j==0): return 1
    else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
[[T(n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 21 2020