%I #87 Apr 12 2022 02:02:03
%S 1,1,1,1,2,1,1,3,3,1,1,4,5,4,1,1,5,8,8,5,1,1,6,12,13,12,6,1,1,7,17,21,
%T 21,17,7,1,1,8,23,33,34,33,23,8,1,1,9,30,50,55,55,50,30,9,1,1,10,38,
%U 73,88,89,88,73,38,10,1,1,11,47,103,138,144,144,138,103,47,11,1
%N Rectangular array defined by T(i,1) = T(1,j) = 1 for i >= 1 and j >= 1; T(i,j) = max(T(i-1,j) + T(i-1,j-1), T(i-1,j-1) + T(i,j-1)) for i >= 2, j >= 2, read by antidiagonals.
%C Antidiagonal sums: A029907.
%C Main diagonal: A001519 (odd-indexed Fibonacci numbers).
%C Next diagonal: A001906 (even-indexed Fibonacci numbers).
%H G. C. Greubel, <a href="/A038792/b038792.txt">Antidiagonals n = 1..50, flattened</a>
%H H. Belbachir and A. Belkhir, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Belbachir/belb2.html">Combinatorial Expressions Involving Fibonacci, Hyperfibonacci, and Incomplete Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.4.3.
%H A. Dil and I. Mezo, <a href="http://dx.doi.org/10.1016/j.amc.2008.10.013">A symmetric algorithm for hyperharmonic and Fibonacci numbers</a>, Appl. Math. Comp. 206 (2008), 942-951; in Eqs. (11), see the incomplete Fibonacci numbers.
%H Piero Filipponi, <a href="https://doi.org/10.1007/BF02845088">Incomplete Fibonacci and Lucas numbers</a>, P. Rend. Circ. Mat. Palermo (Serie II) 45(1) (1996), 37-56; see Table 1 that contains the incomplete Fibonacci numbers.
%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; see Example 4 (appears as a triangular array).
%H A. Pintér and H.M. Srivastava, <a href="https://doi.org/10.1007/BF02844348">Generating functions of the incomplete Fibonacci and Lucas numbers</a>, Rend. Circ. Mat. Palermo (Serie II) 48(3) (1999), 591-596.
%F G.f.: x*y*(1-x*y)/((x*y+x-1)*(x*y+y-1)). - _Mark van Hoeij_, Nov 09 2011
%F From _Petros Hadjicostas_, Sep 02 2019: (Start)
%F Following Dil and Mezo (2008), define the incomplete Fibonacci numbers by F(n,k) = Sum_{s = 0..k} binomial(n-1-s, s) for n >= 1 and 0 <= k <= floor((n-1)/2).
%F Then T(i, j) = F(i+j-1, min(i-1, j-1)) for i,j >= 1.
%F (End)
%e From _Clark Kimberling_, Jun 20 2011: (Start)
%e Northwest corner begins at (i,j) = (1,1):
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, 6, 7, 8, ...
%e 1, 3, 5, 8, 12, 17, 23, 30, ...
%e 1, 4, 8, 13, 21, 33, 50, 73, ...
%e 1, 5, 12, 21, 34, 55, 88, 138, ...
%e 1, 6, 17, 33, 55, 89, 144, 232, ...
%e 1, 7, 23, 50, 88, 144, 233, 377, ...
%e (End)
%e Antidiagonal triangle begins as:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 3, 3, 1;
%e 1, 4, 5, 4, 1;
%e 1, 5, 8, 8, 5, 1;
%e 1, 6, 12, 13, 12, 6, 1;
%e 1, 7, 17, 21, 21, 17, 7, 1;
%e 1, 8, 23, 33, 34, 33, 23, 8, 1;
%e 1, 9, 30, 50, 55, 55, 50, 30, 9, 1;
%p G := x*y*(1-x*y)/((x*y+x-1)*(x*y+y-1)); G := convert(series(G, x=0, 11),polynom):
%p for i from 1 to 10 do series(coeff(G,x,i),y=0,11) od; # _Mark van Hoeij_, Nov 09 2011
%p # second Maple program:
%p G:= x*y*(1-x*y)/((x*y+x-1)*(x*y+y-1)):
%p T:= (i, j)-> coeff(series(coeff(series(G, y, j+1), y, j), x, i+1), x, i):
%p seq(seq(T(i, 1+d-i), i=1..d), d=1..12); # _Alois P. Heinz_, Sep 02 2019
%p # third Maple program:
%p T:= proc(i,j) option remember; `if`(i=1 or j=1, 1,
%p max(T(i-1,j) + T(i-1,j-1), T(i-1,j-1) + T(i,j-1)))
%p end:
%p seq(seq(T(i, 1+d-i), i=1..d), d=1..12); # _Alois P. Heinz_, Sep 02 2019
%t f[i_, 0]:= 1; f[0, i_]:= 1
%t f[i_, j_]:= f[i,j]= Max[f[i-1,j] +f[i-1,j-1], f[i-1,j-1] +f[i,j-1]];
%t T[i_, j_]:= f[i-j, j-1];
%t TableForm[Table[f[i, j], {i,0,7}, {j,0,7}]]
%t Table[T[i, j], {i,10}, {j,i}]//Flatten (* modified by _G. C. Greubel_, Apr 05 2022 *)
%o (Magma)
%o function t(n,k)
%o if k eq 0 or n eq 0 then return 1;
%o else return Max(t(n-1,k-1) + t(n-1,k), t(n-1,k-1) + t(n,k-1));
%o end if; return t;
%o end function;
%o T:= func< n,k | t(n-k, k-1) >;
%o [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 05 2022
%o (SageMath)
%o def t(n,k):
%o if (k==0 or n==0): return 1
%o else: return max(t(n-1,k-1) + t(n-1,k), t(n-1,k-1) + t(n,k-1))
%o def A038792(n,k): return t(n-k, k-1)
%o flatten([[A038792(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Apr 05 2022
%Y Cf. A001519, A001906, A029907.
%Y Cf. A000045, A038730, A134511, A324242.
%Y Main diagonal gives A001519.
%K nonn,tabl,easy
%O 1,5
%A _Clark Kimberling_, May 02 2000
%E New description from _Benoit Cloitre_, Aug 05 2003
%E Updated from pre-2003 triangular format to present rectangular, from _Clark Kimberling_, Jun 20 2011