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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.

%I A055818 #23 Jan 05 2025 19:51:36
%S A055818 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,
%T A055818 144,122,69,27,7,1,1,191,336,328,217,103,35,8,1,1,383,768,848,640,354,
%U A055818 146,44,9,1,1,767,1728,2128,1800,1131,543,199,54,10,1
%N 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.
%H A055818 G. C. Greubel, <a href="/A055818/b055818.txt">Rows n = 0..100 of triangle, flattened</a>
%H A055818 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/40-4/kimberling.pdf">Path-counting and Fibonacci numbers</a>, Fib. Quart. 40 (4) (2002) 328-338, Example 3B.
%e A055818 Rows begins as:
%e A055818   1;
%e A055818   1,  1;
%e A055818   1,  2, 1;
%e A055818   1,  5, 3, 1;
%e A055818   1, 11, 9, 4, 1;
%e A055818   ...
%p A055818 T:= proc(i, j) option remember;
%p A055818       if i=0 or j=0 then 1
%p A055818     else add(add(T(h,m), m=0..j), h=0..i-1)
%p A055818       fi; end:
%p A055818 seq(seq(T(n-k, k), k=0..n), n=0..12); # _G. C. Greubel_, Jan 21 2020
%t A055818 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 *)
%o A055818 (PARI) T(i,j) = if(i==0 || j==0, 1, sum(h=0,i-1, sum(m=0,j, T(h,m) )));
%o A055818 for(n=0,12, for(k=0, n, print1(T(n-k,k), ", "))) \\ _G. C. Greubel_, Jan 21 2020
%o A055818 (Magma)
%o A055818 function T(i,j)
%o A055818   if i eq 0 or j eq 0 then return 1;
%o A055818   else return (&+[(&+[T(h,m): m in [0..j]]): h in [0..i-1]]);
%o A055818   end if; return T; end function;
%o A055818 [T(n-k,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 21 2020
%o A055818 (Sage)
%o A055818 @CachedFunction
%o A055818 def T(i, j):
%o A055818     if (i==0 or j==0): return 1
%o A055818     else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))
%o A055818 [[T(n-k, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 21 2020
%o A055818 (GAP)
%o A055818 T:= function(i,j)
%o A055818     if i=0 or j=0 then return 1;
%o A055818     else return Sum([0..i-1], h-> Sum([0..j], m-> T(h,m) ));
%o A055818     fi; end;
%o A055818 Flat(List([0..12], n-> List([0..n], k-> T(n-k,k) ))); # _G. C. Greubel_, Jan 21 2020
%Y A055818 Cf. A055819, A055820, A055821, A055822, A055823, A055824, A055825, A055826, A055827, A055828, A055829.
%K A055818 nonn,tabl
%O A055818 0,5
%A A055818 _Clark Kimberling_, May 28 2000