A081576 - OEIS

%I #13 Aug 23 2024 22:05:59

%S 0,0,1,0,1,1,0,1,3,2,0,1,5,8,3,0,1,7,20,21,5,0,1,9,38,75,55,8,0,1,11,

%T 62,189,275,144,13,0,1,13,92,387,905,1000,377,21,0,1,15,128,693,2305,

%U 4256,3625,987,34,0,1,17,170,1131,4955,13392,19837,13125,2584,55

%N Square array of binomial transforms of Fibonacci numbers, read by antidiagonals.

%C Array rows are solutions of the recurrence a(n) = (2*k+1)*a(n-1) - A028387(k-1)*a(n-2) where a(0) = 0 and a(1) = 1.

%H G. C. Greubel, <a href="/A081576/b081576.txt">Antidiagonal rows n = 0..50, flattened</a>

%F Rows are successive binomial transforms of F(n).

%F T(n, k) = ( ( (2*n + 1 + sqrt(5))/2 )^k - ( (2*n + 1 - sqrt(5))/2 )^k )/sqrt(5).

%F From _G. C. Greubel_, May 26 2021: (Start)

%F T(n, k) = Sum_{j=0..k} binomial(k,j)*Fibonacci(j)*n^(k-j) with T(0, k) = Fibonacci(k) (square array).

%F T(n, k) = Sum_{j=0..k} binomial(k,j)*Fibonacci(j)*(n-k)^(k-j) (antidiagonal triangle). (End)

%e Square array begins as:

%e   0, 1,  1,   2,    3,    5,     8, ... A000045;

%e   0, 1,  3,   8,   21,   55,   144, ... A001906;

%e   0, 1,  5,  20,   75,  275,  1000, ... A030191;

%e   0, 1,  7,  38,  189,  905,  4256, ... A099453;

%e   0, 1,  9,  62,  387, 2305, 13392, ... A081574;

%e   0, 1, 11,  92,  693, 4955, 34408, ... A081575;

%e   0, 1, 13, 128, 1131, 9455, 76544, ...

%e The antidiagonal triangle begins as:

%e   0;

%e   0, 1;

%e   0, 1,  1;

%e   0, 1,  3,  2;

%e   0, 1,  5,  8,   3;

%e   0, 1,  7, 20,  21,   5;

%e   0, 1,  9, 38,  75,  55,   8;

%e   0, 1, 11, 62, 189, 275, 144, 13;

%t T[n_, k_]:= If[n==0, Fibonacci[k], Sum[Binomial[k, j]*Fibonacci[j]*n^(k-j), {j, 0, k}]]; Table[T[n-k, k], {n,0,12}, {k,0,n}] //Flatten (* _G. C. Greubel_, May 26 2021 *)

%o (Magma)

%o A081576:= func< n,k | (&+[Binomial(k,j)*Fibonacci(j)*(n-k)^(k-j): j in [0..k]]) >;

%o [A081576(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 26 2021

%o (Sage)

%o def A081576(n,k): return sum( binomial(k,j)*fibonacci(j)*(n-k)^(k-j) for j in (0..k) )

%o flatten([[A081576(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 26 2021

%Y Array row n: A000045 (n=0), A001906 (n=1), A030191 (n=2), A099453 (n=3), A081574 (n=4), A081575 (n=5).

%Y Array columns k: A005408 (k=3), A077588 (k=4).

%Y Cf. A028387, A081573, A081574, A081575.

%K easy,nonn,tabl

%O 0,9

%A _Paul Barry_, Mar 22 2003