%I #24 Sep 08 2022 08:45:01

%S 1,1,1,1,2,1,1,3,2,1,1,4,4,2,0,1,5,7,4,1,0,1,6,11,8,3,0,0,1,7,16,15,7,

%T 1,0,0,1,8,22,26,15,4,0,0,0,1,9,29,42,30,11,1,0,0,0,1,10,37,64,56,26,

%U 5,0,0,0,0,1,11,46,93,98,56,16,1,0,0,0,0,1,12,56,130,162,112,42,6,0,0,0,0,0

%N Triangle T read by rows: T(i,0)=1 for i >= 0; T(i,i)=1 for i=0,1,2,3; T(i,i)=0 for i >= 4; T(i,j) = T(i-1,j) + T(i-2,j-1) for 1<=j<=i-1.

%C T(i+j,j) is the number of strings (s(1),...,s(i+1)) of nonnegative integers s(k) such that 0<=s(k)-s(k-1)<=1 for k=2,3,...,i+1 and s(i+1)=j.

%C T(i+j,j) is the number of compositions of j consisting of i parts, all of in {0,1}.

%H G. C. Greubel, <a href="/A055794/b055794.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 1B.

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 3, 2, 1;

%e 1, 4, 4, 2, 0;

%e 1, 5, 7, 4, 1, 0;

%e ...

%e T(7,4) counts the strings 3334, 3344, 3444, 2234, 2334, 2344, 1234.

%e T(7,4) counts the compositions 001, 010, 100, 011, 101, 110, 111.

%p T:= proc(n, k) option remember;

%p if k=0 then 1

%p elif k=n and n<4 then 1

%p elif k=n then 0

%p else T(n-1, k) + T(n-2, k-1)

%p fi; end:

%p seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Jan 25 2020

%t T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==n && n<4, 1, If[k==n, 0, T[n-1, k] + T[n-2, k-1] ]]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 25 2020 *)

%o (PARI) T(n,k) = if(k==0, 1, if(k==n && n<4, 1, if(k==n, 0, T(n-1, k) + T(n-2, k-1) )));

%o for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Jan 25 2020

%o (Magma)

%o function T(n,k)

%o if k eq 0 then return 1;

%o elif k eq n and n lt 4 then return 1;

%o elif k eq n then return 0;

%o else return T(n-1,k) + T(n-2, k-1);

%o end if; return T; end function;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 25 2020

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o if (k==0): return 1

%o elif (k==n and n<4): return 1

%o elif (k==n): return 0

%o else: return T(n-1, k) + T(n-2, k-1)

%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 25 2020

%o (GAP)

%o T:= function(n,k)

%o if k=0 then return 1;

%o elif k=n and n<4 then return 1;

%o elif k=n then return 0;

%o else return T(n-1,k) + T(n-2,k-1);

%o fi; end;

%o Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Jan 25 2020

%Y Row sums: A000204 (Lucas numbers).

%Y Cf. subsequences T(2n+m,n): A000125 (m=0, cake numbers), A055795 (m=1), A027660 (m=2), A055796 (m=3), A055797 (m=4), A055798 (m=5), A055799 (m=6).

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, May 28 2000

%E Typo in definition corrected by _Georg Fischer_, Dec 03 2021