%I #26 Sep 19 2024 12:02:27

%S 1,2,1,3,2,1,5,4,2,1,8,8,5,2,1,13,15,11,6,2,1,21,28,23,14,7,2,1,34,51,

%T 47,32,17,8,2,1,55,92,93,70,42,20,9,2,1,89,164,181,148,97,53,23,10,2,

%U 1,144,290,346,306,217,128,65,26,11,2,1,233,509,653,619,472

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A207610; see Formula section.

%C Column 1: Fibonacci numbers, A000045

%C Column 2: A029907

%C Row sums: A003945.

%C For a discussion and guide to related arrays, see A208510.

%C Subtriangle of the triangle given by (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 25 2012

%F u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = u(n-1,x) + x*v(n-1,x)+1, where u(1,x)=1, v(1,x)=1.

%F T(n,k) = T(n-1,k) + (n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(1,0) = T(2,1) = 1, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k >= n.

%e First five rows:

%e 1;

%e 2, 1;

%e 3, 2, 1;

%e 5, 4, 2, 1;

%e 8, 8, 5, 2, 1;

%e From _Philippe Deléham_, Mar 25 2012: (Start)

%e (0, 2, -1/2, -1/2, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, ...) begins:

%e 1;

%e 0, 1;

%e 0, 2, 1;

%e 0, 3, 2, 1;

%e 0, 5, 4, 2, 1;

%e 0, 8, 8, 5, 2, 1;

%e 0, 13, 15, 11, 6, 2, 1;

%e 0, 21, 28, 23, 14, 7, 2, 1; (End)

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + v[n - 1, x]

%t v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1

%t Table[Factor[u[n, x]], {n, 1, z}]

%t Table[Factor[v[n, x]], {n, 1, z}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A207610 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A207611 *)

%t T[ n_, k_] := Which[k<0 || n<0, 0, n<2, Boole[k<=n] + Boole[k==0&&n==1], True, T[n, k] = T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1] ]; (* _Michael Somos_, Sep 19 2024 *)

%o (Python)

%o from sympy import Poly

%o from sympy.abc import x

%o def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)

%o def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1

%o def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]

%o for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 28 2017

%o (PARI) {T(n, k) = if(k<0 || n<0, 0, n<2, (k<=n) + (k==0 && n==1), T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-2, k-1) )}; /* _Michael Somos_, Sep 19 2024 */

%Y Cf. A207610, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Feb 19 2012