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%I #23 Apr 26 2023 18:53:10
%S 1,1,1,2,2,2,3,6,6,3,5,12,24,12,5,8,25,60,60,25,8,13,48,150,180,150,
%T 48,13,21,91,336,525,525,336,91,21,34,168,728,1344,1750,1344,728,168,
%U 34,55,306,1512,3276,5040,5040,3276,1512,306,55,89,550,3060,7560,13650,16128,13650,7560,3060,550,89
%N A triangle based on A000045 and Pascal's triangle: T(n,m) = Fibonacci(n-m+1) * Fibonacci(m+1) * binomial(n,m).
%C Row sums give A081057.
%H Reinhard Zumkeller, <a href="/A109906/b109906.txt">Rows n = 0..120 of table, flattened</a>
%H Peter McCalla, Asamoah Nkwanta, <a href="https://arxiv.org/abs/1901.07092">Catalan and Motzkin Integral Representations</a>, arXiv:1901.07092 [math.NT], 2019.
%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F T(n,m) = Fibonacci(n-m+1)*Fibonacci(m+1)*binomial(n,m).
%F T(n,k) = A058071(n,k) * A007318(n,k). - _Reinhard Zumkeller_, Aug 15 2013
%e Triangle T(n,k) begins:
%e 1;
%e 1, 1;
%e 2, 2, 2;
%e 3, 6, 6, 3;
%e 5, 12, 24, 12, 5;
%e 8, 25, 60, 60, 25, 8;
%e 13, 48, 150, 180, 150, 48, 13;
%e 21, 91, 336, 525, 525, 336, 91, 21;
%e 34, 168, 728, 1344, 1750, 1344, 728, 168, 34;
%e 55, 306, 1512, 3276, 5040, 5040, 3276, 1512, 306, 55;
%e 89, 550, 3060, 7560, 13650, 16128, 13650, 7560, 3060, 550, 89;
%e ...
%p f:= n-> combinat[fibonacci](n+1):
%p T:= (n, k)-> binomial(n, k)*f(k)*f(n-k):
%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Apr 26 2023
%t Clear[t, n, m] t[n_, m_] := Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]*Binomial[n, m]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]
%o (Haskell)
%o a109906 n k = a109906_tabl !! n !! k
%o a109906_row n = a109906_tabl !! n
%o a109906_tabl = zipWith (zipWith (*)) a058071_tabl a007318_tabl
%o -- _Reinhard Zumkeller_, Aug 15 2013
%Y Cf. A141611, A141617, A000045, A081057.
%Y Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A111006, A114197, A162741, A228074.
%K nonn,tabl
%O 0,4
%A _Roger L. Bagula_ and _Gary W. Adamson_, Aug 24 2008
%E Offset changed by _Reinhard Zumkeller_, Aug 15 2013
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