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%I #25 Apr 13 2020 06:24:12
%S 1,2,3,3,4,11,3,5,26,20,3,6,50,74,29,3,7,85,204,149,38,3,8,133,469,
%T 547,251,47,3,9,196,952,1618,1160,380,56,3,10,276,1764,4110,4234,2124,
%U 536,65,3,11,375,3048,9318,13036,9262,3520,719,74,3,12,495,4983
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A207609; see the Formula section.
%C As triangle T(n,k) with 0<=k<=n and with zeros omitted, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 03 2012
%F u(n,x) = u(n-1,x) + v(n-1,x),
%F v(n,x) = 2x*u(n-1,x) + (x+1)v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Mar 03 2012: (Start)
%F As triangle T(n,k), 0 <= k <= n:
%F T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-1,k) + T(n-2,k-1) with T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k < 0 or if k > n.
%F G.f.: (1-y*x)/(1 - (2+y)*x - (y-1)*x^2).
%F Sum_{k=0..n} T(n,k)*x^k = A000027(n+1), A025192(n), A001077(n), A180038(n) for x = 0, 1, 2, 3 respectively. (End)
%e First five rows:
%e 1;
%e 2;
%e 3, 3;
%e 4, 11, 3;
%e 5, 26, 20, 3;
%e Triangle (2, -1/2, 1/2, 0, 0, 0, 0, ...) DELTA (0, 3/2, -1/2, 0, 0, 0, 0, ...) begins:
%e 1;
%e 2, 0;
%e 3, 3, 0;
%e 4, 11, 3, 0;
%e 5, 26, 20, 3, 0;
%e 6, 50, 74, 29, 3, 0;
%e 7, 85, 204, 149, 38, 3, 0;
%e ... - _Philippe Deléham_, Mar 03 2012
%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_] := 2 x*u[n - 1, x] + (x + 1) v[n - 1, x]
%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[%] (* A207608 *)
%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[%] (* A207609 *)
%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 2*x*u(n - 1, x) + (x + 1)*v(n - 1, x)
%o def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
%o for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 28 2017
%Y Cf. A207609.
%K nonn,tabf
%O 1,2
%A _Clark Kimberling_, Feb 19 2012
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