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A106198 Triangle, columns = successive binomial transforms of Fibonacci numbers.

%I A106198 #13 May 04 2024 09:23:56
%S A106198 1,1,1,2,2,1,3,5,3,1,5,13,10,4,1,8,34,35,17,5,1,13,89,125,75,26,6,1,
%T A106198 21,233,450,338,139,37,7,1,34,610,1625,1541,757,233,50,8,1
%N A106198 Triangle, columns = successive binomial transforms of Fibonacci numbers.
%C A106198 Column 0 = Fibonacci numbers, column 1 = odd-indexed Fibonacci numbers (first binomial transform of 1, 1, 2, 3, 5, ...); column 2 = second binomial transform of Fibonacci numbers, etc.
%H A106198 G. C. Greubel, <a href="/A106198/b106198.txt">Rows n = 0..100 of triangle, flattened</a>
%F A106198 Offset column k = k-th binomial transform of the Fibonacci numbers, given leftmost column = Fibonacci numbers.
%e A106198 First few rows of the triangle are:
%e A106198    1;
%e A106198    1,   1;
%e A106198    2,   2,   1;
%e A106198    3,   5,   3,   1;
%e A106198    5,  13,  10,   4,   1;
%e A106198    8,  34,  35,  17,   5,   1;
%e A106198   13,  89, 125,  75,  26,   6,   1;
%e A106198   21, 233, 450, 338, 139,  37,   7,   1;
%e A106198   ...
%e A106198 Column 2 = A081567, second binomial transform of Fibonacci numbers: 1, 3, 10, 35, 125, ...
%p A106198 with(combinat);
%p A106198 T:= proc(n, k) option remember;
%p A106198       if k=0 then fibonacci(n+1)
%p A106198     else add( binomial(n-k,j)*fibonacci(j+1)*k^(n-k-j), j=0..n-k)
%p A106198       fi; end:
%p A106198 seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Dec 11 2019
%t A106198 Table[If[k==0, Fibonacci[n+1], Sum[Binomial[n-k, j]*Fibonacci[j+1]*k^(n-k-j), {j,0,n-k}]], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 11 2019 *)
%o A106198 (PARI) T(n,k) = if(k==0, fibonacci(n+1), sum(j=0,n-k, binomial(n-k,j)*fibonacci( j+1)*k^(n-k-j)) ); \\ _G. C. Greubel_, Dec 11 2019
%o A106198 (Magma)
%o A106198 function T(n,k)
%o A106198   if k eq 0 then return Fibonacci(n+1);
%o A106198   else return (&+[Binomial(n-k,j)*Fibonacci(j+1)*k^(n-k-j): j in [0..n-k]]);
%o A106198   end if; return T; end function;
%o A106198 [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 11 2019
%o A106198 (Sage)
%o A106198 @CachedFunction
%o A106198 def T(n, k):
%o A106198     if (k==0): return fibonacci(n+1)
%o A106198     else: return sum(binomial(n-k,j)*fibonacci(j+1)*k^(n-k-j) for j in (0..n-k))
%o A106198 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 11 2019
%o A106198 (GAP)
%o A106198 T:= function(n,k)
%o A106198     if k=0 then return Fibonacci(n+1);
%o A106198     else return Sum([0..n-k], j-> Binomial(n-k,j)*Fibonacci(j+1)*k^(n-k-j));
%o A106198     fi; end;
%o A106198 Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Dec 11 2019
%Y A106198 Cf. A000045, A081567, A081568, A081569, A081570.
%K A106198 nonn,tabl
%O A106198 0,4
%A A106198 _Gary W. Adamson_, Apr 24 2005