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A259708 Triangle T(n,k) (0 <= k <= n) giving coefficients of certain polynomials related to Fibonacci numbers.

%I A259708 #29 Jan 05 2025 19:51:40
%S A259708 1,0,1,1,-1,2,0,3,0,3,1,0,14,4,5,0,8,22,60,22,8,1,6,99,244,279,78,13,
%T A259708 0,21,240,1251,2016,1251,240,21,1,25,715,5245,14209,14083,5329,679,34,
%U A259708 0,55,1828,21532,88060,139930,88060,21532,1828,55,1,78,4817,83060,507398,1218920,1219382,507068,83225,4762,89
%N A259708 Triangle T(n,k) (0 <= k <= n) giving coefficients of certain polynomials related to Fibonacci numbers.
%C A259708 The terms are the coefficients of the polynomials given by r_0(x) = 1; r_1(x) = x; r_(n+1) = (n+1)*x*r_n(x) + x*(1-x)*(r_n)'(x) + (1 - x)^2*r_(n-1)(x). [Carlitz, (1.6)]. Note: Carlitz wrongly states r_1(x) = 1. - _Eric M. Schmidt_, Jul 10 2015
%H A259708 Eric M. Schmidt, <a href="/A259708/b259708.txt">Rows n = 0..50, flattened</a>
%H A259708 L. Carlitz, <a href="/A006502/a006502.pdf">Some polynomials related to Fibonacci and Eulerian numbers</a>, Fib. Quart., 16 (1978), 217. (Annotated scanned copy)
%H A259708 L. Carlitz, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/16-3/carlitz1.pdf">Some polynomials related to Fibonacci and Eulerian numbers</a>, Fib. Quart., 16 (1978), 216-226.
%F A259708 T(0,0) = 1; T(n+1,k) = (n-k+2)*T(n,k-1) + k*T(n,k) + T(n-1,k) - 2*T(n-1,k-1) + T(n-1,k-2), where we put T(n,k) = 0 if n < 0 or k < 0. As special cases, T(n,n) = Fibonacci(n+1) and T(n,0) = 1 (n even) or 0 (n odd). - Rewritten by _Eric M. Schmidt_, Jul 10 2015
%e A259708 Triangle begins:
%e A259708 1,
%e A259708 0,1,
%e A259708 1,-1,2,
%e A259708 0,3,0,3,
%e A259708 1,0,14,4,5,
%e A259708 0,8,22,60,22,8,
%e A259708 1,6,99,244,279,78,13,
%e A259708 0,21,240,1251,2016,1251,240,21,
%e A259708 ...
%p A259708 A259708  := proc(n,k)
%p A259708     if k < 0 or k > n then
%p A259708         0;
%p A259708     elif k =0 and n =0 then
%p A259708         1;
%p A259708     else
%p A259708         (n-k+1)*procname(n-1,k-1)+k*procname(n-1,k)+procname(n-2,k)-2*procname(n-2,k-1) + procname(n-2,k-2) ;
%p A259708     end if ;
%p A259708 end proc: # _R. J. Mathar_, Jun 18 2019
%t A259708 T[n_, k_] := T[n, k] = If[k < 0 || k > n, 0, If[k == 0 && n == 0, 1, (n - k + 1) T[n - 1, k - 1] + k T[n - 1, k] + T[n - 2, k] - 2 T[n - 2, k - 1] + T[n - 2, k - 2]]];
%t A259708 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 30 2020 *)
%o A259708 (Sage)
%o A259708 @CachedFunction
%o A259708 def T(n,k) :
%o A259708     if n < 0 or k < 0 : return 0
%o A259708     if n == 0 and k == 0 : return 1
%o A259708     return (n-k+1)*T(n-1,k-1) + k*T(n-1,k) + T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2)
%o A259708 # _Eric M. Schmidt_, Jul 10 2015
%Y A259708 Diagonals include A000045, A259709, A006502.
%Y A259708 Cf. A000142 (row sums).
%K A259708 sign,tabl,easy
%O A259708 0,6
%A A259708 _N. J. A. Sloane_, Jul 05 2015
%E A259708 More terms from and name revised by _Eric M. Schmidt_, Jul 10 2015