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%I #35 Jan 14 2024 12:39:57
%S 1,1,1,1,2,2,1,3,3,3,1,4,5,5,5,1,5,8,8,8,8,1,6,12,13,13,13,13,1,7,17,
%T 21,21,21,21,21,1,8,23,33,34,34,34,34,34,1,9,30,50,55,55,55,55,55,55,
%U 1,10,38,73,88,89,89,89,89,89,89,1,11,47,103,138,144,144,144,144,144,144,144,1,12,57
%N Array read by antidiagonals: T(0,k) = A000045(k+1) for k >= 0. T(n,0) = 1 for n >= 0; thereafter T(n,k) = T(n-1,k-1)+T(n-1,k) for n, k >= 1.
%C This is another "Fibonacci array".
%H Robert Israel, <a href="/A294453/b294453.txt">Table of n, a(n) for n = 0..10010</a> (antidiagonals 0 to 140, flattened)
%H Richard L. Ollerton and Anthony G. Shannon, <a href="http://www.fq.math.ca/Scanned/36-2/ollerton.pdf">Some properties of generalized Pascal squares and triangles</a>, Fib. Q., 36 (1998), 98-109. See Table 6.
%F G.f. as array: 1/((1-x-x*y)*(1-y-y^2)). - _Robert Israel_, Nov 22 2017
%F T(n,m) = Sum_{k=0..m} binomial(n+m-k,k). - _Vladimir Kruchinin_, Jan 14 2024
%e The array begins:
%e 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
%e 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
%e 1, 3, 5, 8, 13, 21, 34, 55, 89, ...
%e 1, 4, 8, 13, 21, 34, 55, 89, 144, ...
%e 1, 5, 12, 21, 34, 55, 89, 144, 233, ...
%e 1, 6, 17, 33, 55, 89, 144, 233, 377, ...
%e 1, 7, 23, 50, 88, 144, 233, 377, 610, ...
%e 1, 8, 30, 73, 138, 232, 377, 610, 987, ...
%e 1, 9, 38, 103, 211, 370, 609, 987, 1597, ...
%e ...
%e The first few antidiagonals are:
%e 1,
%e 1, 1,
%e 1, 2, 2,
%e 1, 3, 3, 3,
%e 1, 4, 5, 5, 5,
%e 1, 5, 8, 8, 8, 8,
%e 1, 6, 12, 13, 13, 13, 13,
%e 1, 7, 17, 21, 21, 21, 21, 21,
%e ...
%p A294453:= proc(n,k) option remember;
%p if n = 0 then combinat:-fibonacci(k+1)
%p elif k = 0 then 1
%p else procname(n-1,k-1)+procname(n-1,k)
%p fi
%p end proc:
%p seq(seq(A294453(m-k,k),k=0..m),m=0..10); # _Robert Israel_, Nov 22 2017
%t T[0, k_] := Fibonacci[k + 1];
%t T[_, 0] = 1;
%t T[n_, k_] := T[n, k] = T[n - 1, k - 1] + T[n - 1, k];
%t Table[T[n - k, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 16 2020 *)
%Y Cf. A000045, A001629 (sums of antidiagonals).
%K nonn,tabl
%O 0,5
%A _N. J. A. Sloane_, Nov 22 2017
%E Better definition and more terms from _Robert Israel_, Nov 22 2017
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