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A109906 A triangle based on A000045 and Pascal's triangle: T(n,m) = Fibonacci(n-m+1) * Fibonacci(m+1) * binomial(n,m).

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