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A228074 A Fibonacci-Pascal triangle read by rows: T(n,0) = Fibonacci(n), T(n,n) = n and for n > 0: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n.

%I A228074 #30 Aug 30 2022 14:13:12
%S A228074 0,1,1,1,2,2,2,3,4,3,3,5,7,7,4,5,8,12,14,11,5,8,13,20,26,25,16,6,13,
%T A228074 21,33,46,51,41,22,7,21,34,54,79,97,92,63,29,8,34,55,88,133,176,189,
%U A228074 155,92,37,9,55,89,143,221,309,365,344,247,129,46,10
%N A228074 A Fibonacci-Pascal triangle read by rows: T(n,0) = Fibonacci(n), T(n,n) = n and for n > 0: T(n,k) = T(n-1,k-1) + T(n-1,k), 0 < k < n.
%C A228074 Sum of n-th row is 2^(n+1) - F(n+1) - 1 = A228078(n+1). - _Greg Dresden_ and _Sadek Mohammed_, Aug 30 2022
%H A228074 Reinhard Zumkeller, <a href="/A228074/b228074.txt">Rows n = 0..120 of table, flattened</a>
%H A228074 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%e A228074 .    0:                                 0
%e A228074 .    1:                               1   1
%e A228074 .    2:                             1   2   2
%e A228074 .    3:                          2    3    4   3
%e A228074 .    4:                       3    5    7    7   4
%e A228074 .    5:                     5    8   12   14   11   5
%e A228074 .    6:                  8   13   20   26   25   16   6
%e A228074 .    7:               13   21   33   46   51   41   22   7
%e A228074 .    8:            21   34   54   79   97   92   63   29   8
%e A228074 .    9:          34   55   88  133  176  189  155   92   37   9
%e A228074 .   10:       55   89  143  221  309  365  344  247  129   46  10
%e A228074 .   11:     89  144  232  364  530  674  709  591  376  175  56   11
%e A228074 .   12:  144 233  376  596  894 1204 1383 1300  967  551  231  67   12 .
%p A228074 with(combinat);
%p A228074 T:= proc (n, k) option remember;
%p A228074 if k = 0 then fibonacci(n)
%p A228074 elif k = n then n
%p A228074 else T(n-1, k-1) + T(n-1, k)
%p A228074 end if
%p A228074 end proc;
%p A228074 seq(seq(T(n, k), k = 0..n), n = 0..12); # _G. C. Greubel_, Sep 05 2019
%t A228074 T[n_, k_]:= T[n, k]= If[k==0, Fibonacci[n], If[k==n, n, T[n-1, k-1] + T[n -1, k]]]; Table[T[n, k], {n,0,12}, {k,0,n}] (* _G. C. Greubel_, Sep 05 2019 *)
%o A228074 (Haskell)
%o A228074 a228074 n k = a228074_tabl !! n !! k
%o A228074 a228074_row n = a228074_tabl !! n
%o A228074 a228074_tabl = map fst $ iterate
%o A228074    (\(u:_, vs) -> (vs, zipWith (+) ([u] ++ vs) (vs ++ [1]))) ([0], [1,1])
%o A228074 (PARI) T(n,k) = if(k==0, fibonacci(n), if(k==n, n, T(n-1, k-1) + T(n-1, k)));
%o A228074 for(n=0, 12, for(k=0, n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Sep 05 2019
%o A228074 (Sage)
%o A228074 def T(n, k):
%o A228074     if (k==0): return fibonacci(n)
%o A228074     elif (k==n): return n
%o A228074     else: return T(n-1, k) + T(n-1, k-1)
%o A228074 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Sep 05 2019
%o A228074 (GAP)
%o A228074 T:= function(n,k)
%o A228074     if k=0 then return Fibonacci(n);
%o A228074     elif k=n then return n;
%o A228074     else return T(n-1,k-1) + T(n-1,k);
%o A228074     fi;
%o A228074   end;
%o A228074 Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Sep 05 2019
%Y A228074 Cf. A000045 (left edge), A001477 (right edge), A228078 (row sums), A027988 (maxima per row);
%Y A228074 diagonals T(*,k): A000045, A000071, A001924, A014162, A014166, A053739, A053295, A053296, A053308, A053309;
%Y A228074 diagonals T(k,*): A001477, A001245, A004006, A027927, A027928, A027929, A027930, A027931, A027932, A027933;
%Y A228074 some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A111006, A114197, A162741.
%K A228074 nonn,tabl,look
%O A228074 0,5
%A A228074 _Reinhard Zumkeller_, Aug 15 2013