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 A259708 Triangle T(n,k) (0 <= k <= n) giving coefficients of certain polynomials related to Fibonacci numbers. 3
 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, 0, 21, 240, 1251, 2016, 1251, 240, 21, 1, 25, 715, 5245, 14209, 14083, 5329, 679, 34, 0, 55, 1828, 21532, 88060, 139930, 88060, 21532, 1828, 55, 1, 78, 4817, 83060, 507398, 1218920, 1219382, 507068, 83225, 4762, 89 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS 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 LINKS Eric M. Schmidt, Rows n = 0..50, flattened L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, Fib. Quart., 16 (1978), 217. (Annotated scanned copy) L. Carlitz, Some polynomials related to Fibonacci and Eulerian numbers, Fib. Quart., 16 (1978), 216-226. FORMULA 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 EXAMPLE Triangle begins: 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, 0,21,240,1251,2016,1251,240,21, ... MAPLE A259708  := proc(n, k)     if k < 0 or k > n then         0;     elif k =0 and n =0 then         1;     else         (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) ;     end if ; end proc: # R. J. Mathar, Jun 18 2019 MATHEMATICA 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]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 30 2020 *) PROG (Sage) @CachedFunction def T(n, k) :     if n < 0 or k < 0 : return 0     if n == 0 and k == 0 : return 1     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) # Eric M. Schmidt, Jul 10 2015 CROSSREFS Diagonals include A000045, A259709, A006502. Cf. A000142 (row sums). Sequence in context: A213177 A265017 A035376 * A029220 A249901 A253274 Adjacent sequences:  A259705 A259706 A259707 * A259709 A259710 A259711 KEYWORD sign,tabl,easy AUTHOR N. J. A. Sloane, Jul 05 2015 EXTENSIONS More terms from and name revised by Eric M. Schmidt, Jul 10 2015 STATUS approved

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Last modified August 8 23:02 EDT 2020. Contains 336300 sequences. (Running on oeis4.)