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A207609 Triangle of coefficients of polynomials v(n,x) jointly generated with A207608; see Formula section.

%I A207609 #20 Apr 13 2020 09:32:25
%S A207609 1,1,3,1,8,3,1,15,17,3,1,24,54,26,3,1,35,130,120,35,3,1,48,265,398,
%T A207609 213,44,3,1,63,483,1071,909,333,53,3,1,80,812,2492,3074,1744,480,62,3,
%U A207609 1,99,1284,5208,8802,7138,2984,654,71,3,1,120,1935,10020,22230,24408,14370,4710,855,80,3
%N A207609 Triangle of coefficients of polynomials v(n,x) jointly generated with A207608; see Formula section.
%C A207609 Subtriangle of the triangle given by (1, 0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 03 2012
%F A207609 u(n,x)=u(n-1,x)+v(n-1,x),
%F A207609 v(n,x)=2x*u(n-1,x)+(x+1)v(n-1,x),
%F A207609 where u(1,x)=1, v(1,x)=1.
%F A207609 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) - T(n-2,k), n>2. - _Philippe Deléham_, Mar 03 2012
%F A207609 Sum_{k, 0<=k<=n, n>=1} T(n,k)*x^k = A000012(n), A052156(n-1), A048876(n-1) for x = 0, 1, 2 respectively. - _Philippe Deléham_, Mar 03 2012
%F A207609 G.f.: -(1-x+2*x*y)*x*y/(-1+2*x+x*y+x^2*y-x^2). - _R. J. Mathar_, Aug 11 2015
%e A207609 First five rows:
%e A207609 1
%e A207609 1...3
%e A207609 1...8....3
%e A207609 1...15...17...3
%e A207609 1...24...54...26...3
%e A207609 Triangle (1, 0, 2/3, 1/3, 0, 0, 0, ...) DELTA (0, 3, -2, 0, 0, 0, ...) begins :
%e A207609 1
%e A207609 1, 0
%e A207609 1, 3, 0
%e A207609 1, 8, 3, 0
%e A207609 1, 15, 17, 3, 0
%e A207609 1, 24, 54, 26, 3, 0
%e A207609 1, 35, 130, 120, 35, 3, 0
%t A207609 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207609 u[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207609 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1) v[n - 1, x]
%t A207609 Table[Factor[u[n, x]], {n, 1, z}]
%t A207609 Table[Factor[v[n, x]], {n, 1, z}]
%t A207609 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207609 TableForm[cu]
%t A207609 Flatten[%]    (* A207608 *)
%t A207609 Table[Expand[v[n, x]], {n, 1, z}]
%t A207609 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207609 TableForm[cv]
%t A207609 Flatten[%]    (* A207609 *)
%o A207609 (Python)
%o A207609 from sympy import Poly
%o A207609 from sympy.abc import x
%o A207609 def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
%o A207609 def v(n, x): return 1 if n==1 else 2*x*u(n - 1, x) + (x + 1)*v(n - 1, x)
%o A207609 def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
%o A207609 for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 28 2017
%Y A207609 Cf. A207608.
%K A207609 nonn,tabl
%O A207609 1,3
%A A207609 _Clark Kimberling_, Feb 19 2012