This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A081576 #13 Aug 23 2024 22:05:59
%S A081576 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 A081576 62,189,275,144,13,0,1,13,92,387,905,1000,377,21,0,1,15,128,693,2305,
%U A081576 4256,3625,987,34,0,1,17,170,1131,4955,13392,19837,13125,2584,55
%N A081576 Square array of binomial transforms of Fibonacci numbers, read by antidiagonals.
%C A081576 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 A081576 G. C. Greubel, <a href="/A081576/b081576.txt">Antidiagonal rows n = 0..50, flattened</a>
%F A081576 Rows are successive binomial transforms of F(n).
%F A081576 T(n, k) = ( ( (2*n + 1 + sqrt(5))/2 )^k - ( (2*n + 1 - sqrt(5))/2 )^k )/sqrt(5).
%F A081576 From _G. C. Greubel_, May 26 2021: (Start)
%F A081576 T(n, k) = Sum_{j=0..k} binomial(k,j)*Fibonacci(j)*n^(k-j) with T(0, k) = Fibonacci(k) (square array).
%F A081576 T(n, k) = Sum_{j=0..k} binomial(k,j)*Fibonacci(j)*(n-k)^(k-j) (antidiagonal triangle). (End)
%e A081576 Square array begins as:
%e A081576 0, 1, 1, 2, 3, 5, 8, ... A000045;
%e A081576 0, 1, 3, 8, 21, 55, 144, ... A001906;
%e A081576 0, 1, 5, 20, 75, 275, 1000, ... A030191;
%e A081576 0, 1, 7, 38, 189, 905, 4256, ... A099453;
%e A081576 0, 1, 9, 62, 387, 2305, 13392, ... A081574;
%e A081576 0, 1, 11, 92, 693, 4955, 34408, ... A081575;
%e A081576 0, 1, 13, 128, 1131, 9455, 76544, ...
%e A081576 The antidiagonal triangle begins as:
%e A081576 0;
%e A081576 0, 1;
%e A081576 0, 1, 1;
%e A081576 0, 1, 3, 2;
%e A081576 0, 1, 5, 8, 3;
%e A081576 0, 1, 7, 20, 21, 5;
%e A081576 0, 1, 9, 38, 75, 55, 8;
%e A081576 0, 1, 11, 62, 189, 275, 144, 13;
%t A081576 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 A081576 (Magma)
%o A081576 A081576:= func< n,k | (&+[Binomial(k,j)*Fibonacci(j)*(n-k)^(k-j): j in [0..k]]) >;
%o A081576 [A081576(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 26 2021
%o A081576 (Sage)
%o A081576 def A081576(n,k): return sum( binomial(k,j)*fibonacci(j)*(n-k)^(k-j) for j in (0..k) )
%o A081576 flatten([[A081576(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 26 2021
%Y A081576 Array row n: A000045 (n=0), A001906 (n=1), A030191 (n=2), A099453 (n=3), A081574 (n=4), A081575 (n=5).
%Y A081576 Array columns k: A005408 (k=3), A077588 (k=4).
%Y A081576 Cf. A028387, A081573, A081574, A081575.
%K A081576 easy,nonn,tabl
%O A081576 0,9
%A A081576 _Paul Barry_, Mar 22 2003