A055818 - OEIS

%I #22 Sep 08 2022 08:45:01

%S 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 144,122,69,27,7,1,1,191,336,328,217,103,35,8,1,1,383,768,848,640,354,

%U 146,44,9,1,1,767,1728,2128,1800,1131,543,199,54,10,1

%N 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 G. C. Greubel, <a href="/A055818/b055818.txt">Rows n = 0..100 of triangle, flattened</a>

%H Clark Kimberling, <a href="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 Rows begins as:

%e   1;

%e   1,  1;

%e   1,  2, 1;

%e   1,  5, 3, 1;

%e   1, 11, 9, 4, 1;

%e   ...

%p T:= proc(i, j) option remember;

%p       if i=0 or j=0 then 1

%p     else add(add(T(h,m), m=0..j), h=0..i-1)

%p       fi; end:

%p seq(seq(T(n-k, k), k=0..n), n=0..12); # _G. C. Greubel_, Jan 21 2020

%t 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 (PARI) T(i,j) = if(i==0 || j==0, 1, sum(h=0,i-1, sum(m=0,j, T(h,m) )));

%o for(n=0,12, for(k=0, n, print1(T(n-k,k), ", "))) \\ _G. C. Greubel_, Jan 21 2020

%o (Magma)

%o function T(i,j)

%o   if i eq 0 or j eq 0 then return 1;

%o   else return (&+[(&+[T(h,m): m in [0..j]]): h in [0..i-1]]);

%o   end if; return T; end function;

%o [T(n-k,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 21 2020

%o (Sage)

%o @CachedFunction

%o def T(i, j):

%o     if (i==0 or j==0): return 1

%o     else: return sum(sum(T(h,m) for m in (0..j)) for h in (0..i-1))

%o [[T(n-k, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 21 2020

%o (GAP)

%o T:= function(i,j)

%o     if i=0 or j=0 then return 1;

%o     else return Sum([0..i-1], h-> Sum([0..j], m-> T(h,m) ));

%o     fi; end;

%o Flat(List([0..12], n-> List([0..n], k-> T(n-k,k) ))); # _G. C. Greubel_, Jan 21 2020

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

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, May 28 2000