cp's OEIS Frontend

A081572 Square array of binomial transforms of Fibonacci numbers, read by ascending antidiagonals.

%I A081572 #19 Sep 08 2022 08:45:09
%S A081572 1,1,1,1,2,2,1,3,5,3,1,4,10,13,5,1,5,17,35,34,8,1,6,26,75,125,89,13,1,
%T A081572 7,37,139,338,450,233,21,1,8,50,233,757,1541,1625,610,34,1,9,65,363,
%U A081572 1490,4172,7069,5875,1597,55,1,10,82,535,2669,9633,23165,32532,21250,4181,89
%N A081572 Square array of binomial transforms of Fibonacci numbers, read by ascending antidiagonals.
%C A081572 Array rows are solutions of the recurrence a(n) = (2*k+1)*a(n-1) - A028387(k-1)*a(n-2), where a(0) = 1 and a(1) = k+1.
%H A081572 G. C. Greubel, <a href="/A081572/b081572.txt">Antidiagonal rows n = 0..50, flattened</a>
%F A081572 Rows are successive binomial transforms of F(n+1).
%F A081572 T(n, k) = ((5+sqrt(5))/10)*( (2*n + 1 + sqrt(5))/2)^k + ((5-sqrt(5)/10)*( 2*n + 1 - sqrt(5))/2 )^k.
%F A081572 From _G. C. Greubel_, May 27 2021: (Start)
%F A081572 T(n, k) = Sum_{j=0..k} binomial(k,j)*n^(k-j)*Fibonacci(j+1) (square array).
%F A081572 T(n, k) = Sum_{j=0..k} binomial(k,j)*(n-k)^(k-j)*Fibonacci(j+1) (antidiagonal triangle). (End)
%e A081572 The array rows begins as:
%e A081572   1, 1,  2,   3,    5,     8,     13, ... A000045;
%e A081572   1, 2,  5,  13,   34,    89,    233, ... A001519;
%e A081572   1, 3, 10,  35,  125,   450,   1625, ... A081567;
%e A081572   1, 4, 17,  75,  338,  1541,   7069, ... A081568;
%e A081572   1, 5, 26, 139,  757,  4172,  23165, ... A081569;
%e A081572   1, 6, 37, 233, 1490,  9633,  62753, ... A081570;
%e A081572   1, 7, 50, 363, 2669, 19814, 148153, ... A081571;
%e A081572 Antidiagonal triangle begins as:
%e A081572   1;
%e A081572   1, 1;
%e A081572   1, 2,  2;
%e A081572   1, 3,  5,   3;
%e A081572   1, 4, 10,  13,   5;
%e A081572   1, 5, 17,  35,  34,    8;
%e A081572   1, 6, 26,  75, 125,   89,   13;
%e A081572   1, 7, 37, 139, 338,  450,  233,  21;
%e A081572   1, 8, 50, 233, 757, 1541, 1625, 610, 34;
%t A081572 T[n_, k_]:= If[n==0, Fibonacci[k+1], Sum[Binomial[k, j]*Fibonacci[j+1]*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 A081572 (Magma)
%o A081572 A081572:= func< n,k | (&+[Binomial(k,j)*Fibonacci(j+1)*(n-k)^(k-j): j in [0..k]]) >;
%o A081572 [A081572(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 27 2021
%o A081572 (Sage)
%o A081572 def A081572(n,k): return sum( binomial(k,j)*fibonacci(j+1)*(n-k)^(k-j) for j in (0..k) )
%o A081572 flatten([[A081572(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 27 2021
%Y A081572 Array row n: A000045 (n=0), A001519 (n=1), A081567 (n=2), A081568 (n=3), A081569 (n=4), A081570 (n=5), A081571 (n=6).
%Y A081572 Array column k: A000027 (k=1), A002522 (k=2).
%Y A081572 Different from A073133.
%Y A081572 Cf. A028387.
%K A081572 easy,nonn,tabl
%O A081572 0,5
%A A081572 _Paul Barry_, Mar 22 2003