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%I #22 Apr 07 2021 19:59:29
%S 1,2,1,5,5,1,13,20,9,1,34,71,51,14,1,89,235,233,105,20,1,233,744,942,
%T 594,190,27,1,610,2285,3522,2860,1295,315,35,1,1597,6865,12473,12402,
%U 7285,2534,490,44,1,4181,20284,42447,49963,36122,16407,4578,726,54,1
%N Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).
%C Companion triangle (odd-indexed members) A060921.
%H G. C. Greubel, <a href="/A060920/b060920.txt">Rows n = 0..50 of the triangle, flattened</a>
%H Yidong Sun, <a href="http://www.fq.math.ca/Papers1/43-4/paper43-4-10b.pdf">Numerical Triangles and Several Classical Sequences</a>, Fib. Quart. 43, no. 4, Nov. 2005, pp. 359-370.
%F T(n, k) = A037027(2*n-k, k).
%F T(n, k) = ((2*(n-k) + 1)*A060921(n-1, k-1) + 4*n*T(n-1, k-1))/(5*k), n >= k >= 1.
%F T(n, 0) = F(n)^2 + F(n+1)^2 = A001519(n), with the Fibonacci numbers F(n) = A000045(n).
%F Sum_{k=0..n} T(n, k) = (2^(2*n + 1) + 1)/3 = A007583(n).
%F G.f. for column m >= 0: x^m*pFe(m+1, x)/(1-3*x+x^2)^(m+1), where pFe(n, x) := Sum_{m=0..n} A061176(n, m)*x^m (row polynomials of signed triangle A061176).
%F G.f.: (1-x*(1+y))/(1 - (3+2*y)*x + (1+y)^2*x^2). - _Vladeta Jovovic_, Oct 11 2003
%e Triangle begins as:
%e 1;
%e 2, 1;
%e 5, 5, 1;
%e 13, 20, 9, 1;
%e 34, 71, 51, 14, 1;
%e 89, 235, 233, 105, 20, 1;
%e 233, 744, 942, 594, 190, 27, 1;
%e 610, 2285, 3522, 2860, 1295, 315, 35, 1;
%e 1597, 6865, 12473, 12402, 7285, 2534, 490, 44, 1;
%e 4181, 20284, 42447, 49963, 36122, 16407, 4578, 726, 54, 1;
%e 10946, 59155, 140109, 190570, 163730, 91959, 33705, 7776, 1035, 65, 1;
%t A060920[n_, k_]:= Sum[Binomial[2*n-k-j, j]*Binomial[2*n-k-2*j, k], {j,0,2*n-k}];
%t Table[A060920[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 06 2021 *)
%o (Magma)
%o A060920:= func< n,k | (&+[Binomial(2*n-k-j, j)*Binomial(2*n-k-2*j, k): j in [0..2*n-k]]) >;
%o [A060920(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 06 2021
%o (Sage)
%o def A060920(n,k): return sum(binomial(2*n-k-j, j)*binomial(2*n-k-2*j, k) for j in (0..2*n-k))
%o flatten([[A060920(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 06 2021
%Y Column sequences: A001519 (k=0), A054444 (k=1), A061178 (k=2), A061179 (k=3), A061180 (k=4), A061181 (k=5).
%Y Cf. A000045, A001519, A007583, A037027, A060921, A061176.
%K nonn,easy,tabl
%O 0,2
%A _Wolfdieter Lang_, Apr 20 2001
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