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%I #53 Apr 06 2022 08:49:57
%S 1,1,2,1,4,5,1,6,12,13,1,8,23,33,34,1,10,38,73,88,89,1,12,57,141,211,
%T 232,233,1,14,80,245,455,581,609,610,1,16,107,393,888,1350,1560,1596,
%U 1597,1,18,138,593,1594,2881,3805,4135,4180,4181
%N Path-counting triangular array T(i,j), read by rows, obtained from array t in A038792 by T(i,j) = t(2*i-j, j) (for i >= 1 and 1 <= j <= i).
%H Alois P. Heinz, <a href="/A038730/b038730.txt">Rows n = 1..200, 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 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 T(n, n) = A001519(n) for n >= 1 (odd-indexed Fibonacci numbers).
%F From _Petros Hadjicostas_, Sep 03 2019: (Start)
%F Following Dil and Mezo (2008, p. 944), 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). Then T(i, j) = F(2*i-1, j-1) for 1 <= j <= i.
%F G.f. for column j: Define g(t,j) = ((1+t)^j * (1+t-t^2) + (1-t)^j * (1-t-t^2))/2, which is a function of t^2. Then the g.f. for column j is Sum_{i >= j} T(i,j)*x^i = x^j * (Fibonacci(2*j-1) * (1-x)^(j+1) + Fibonacci(2*j-2) * x * (1-x)^j - x * g(sqrt(x), j)) / ((1-x)^j * (1-3*x+x^2)). This follows from the results in Pintér and Srivastava (1999).
%F (End)
%e Triangle T(i,j) begins as follows:
%e 1;
%e 1, 2;
%e 1, 4, 5;
%e 1, 6, 12, 13;
%e 1, 8, 23, 33, 34;
%e 1, 10, 38, 73, 88, 89;
%e 1, 12, 57, 141, 211, 232, 233;
%e ... [edited by _Petros Hadjicostas_, Sep 02 2019]
%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 T:= (i, j)-> t(2*i-j, j):
%p seq(seq(T(i, j), j=1..i), i=1..10); # _Alois P. Heinz_, Sep 02 2019
%t T[i_, j_]:= Sum[Binomial[2i-k-2, k], {k,0,j-1}];
%t Table[T[i, j], {i, 1, 10}, {j, 1, i}] // Flatten (* _Jean-François Alcover_, Dec 06 2019 *)
%o (Magma) [(&+[Binomial(2*n-j-2, j): j in [0..k-1]]): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Apr 05 2022
%o (SageMath)
%o def A038730(n,k): return sum( binomial(2*n-j-2, j) for j in (0..k-1))
%o flatten([[A038730(n,k) for k in (1..n)] for n in (1..12)]) # _G. C. Greubel_, Apr 05 2022
%Y Cf. A001519, A038792, A134511, A324242.
%K nonn,tabl
%O 1,3
%A _Clark Kimberling_, May 02 2000
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