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