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Triangle, columns = successive binomial transforms of Fibonacci numbers.
1

%I #13 May 04 2024 09:23:56

%S 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 21,233,450,338,139,37,7,1,34,610,1625,1541,757,233,50,8,1

%N Triangle, columns = successive binomial transforms of Fibonacci numbers.

%C 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 G. C. Greubel, <a href="/A106198/b106198.txt">Rows n = 0..100 of triangle, flattened</a>

%F Offset column k = k-th binomial transform of the Fibonacci numbers, given leftmost column = Fibonacci numbers.

%e First few rows of the triangle are:

%e 1;

%e 1, 1;

%e 2, 2, 1;

%e 3, 5, 3, 1;

%e 5, 13, 10, 4, 1;

%e 8, 34, 35, 17, 5, 1;

%e 13, 89, 125, 75, 26, 6, 1;

%e 21, 233, 450, 338, 139, 37, 7, 1;

%e ...

%e Column 2 = A081567, second binomial transform of Fibonacci numbers: 1, 3, 10, 35, 125, ...

%p with(combinat);

%p T:= proc(n, k) option remember;

%p if k=0 then fibonacci(n+1)

%p else add( binomial(n-k,j)*fibonacci(j+1)*k^(n-k-j), j=0..n-k)

%p fi; end:

%p seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Dec 11 2019

%t 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 (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 (Magma)

%o function T(n,k)

%o if k eq 0 then return Fibonacci(n+1);

%o else return (&+[Binomial(n-k,j)*Fibonacci(j+1)*k^(n-k-j): j in [0..n-k]]);

%o end if; return T; end function;

%o [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 11 2019

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o if (k==0): return fibonacci(n+1)

%o else: return sum(binomial(n-k,j)*fibonacci(j+1)*k^(n-k-j) for j in (0..n-k))

%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 11 2019

%o (GAP)

%o T:= function(n,k)

%o if k=0 then return Fibonacci(n+1);

%o else return Sum([0..n-k], j-> Binomial(n-k,j)*Fibonacci(j+1)*k^(n-k-j));

%o fi; end;

%o Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Dec 11 2019

%Y Cf. A000045, A081567, A081568, A081569, A081570.

%K nonn,tabl

%O 0,4

%A _Gary W. Adamson_, Apr 24 2005