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A210565 Triangle of coefficients of polynomials u(n,x) jointly generated with A210595; see the Formula section.

%I A210565 #12 May 25 2021 01:37:47
%S A210565 1,2,2,3,5,3,4,9,10,5,5,14,22,20,8,6,20,40,51,38,13,7,27,65,105,111,
%T A210565 71,21,8,35,98,190,256,233,130,34,9,44,140,315,511,594,474,235,55,10,
%U A210565 54,192,490,924,1295,1324,942,420,89,11,65,255,726,1554,2534,3130,2860,1836,744,144
%N A210565 Triangle of coefficients of polynomials u(n,x) jointly generated with A210595; see the Formula section.
%C A210565 Row n starts with n and ends with F(n+1), where F=A000045 (Fibonacci numbers).
%C A210565 Row sums: A005409.
%C A210565 Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0, ...
%C A210565 For a discussion and guide to related arrays, see A208510.
%H A210565 G. C. Greubel, <a href="/A210565/b210565.txt">Rows n = 1..30 of the triangle, flattened</a>
%F A210565 u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x) + 1,
%F A210565 v(n,x) = x*u(n-1,x) + v(n-1,x) + 1,
%F A210565 where u(1,x) = 1, v(1,x) = 1.
%F A210565 T(n, k) = [x^k]( u(n, x) ), where u(n, x) = (1+x)*u(n-1,x) + x^2*u(n-2,x) + 1 + x, u(1, x) = 1, and u(2, x) = 2 + 2*x. - _G. C. Greubel_, May 24 2021
%e A210565 First five rows:
%e A210565   1;
%e A210565   2,  2;
%e A210565   3,  5,  3;
%e A210565   4,  9, 10,  5;
%e A210565   5, 14, 22, 20, 8;
%e A210565 First three polynomials u(n,x):
%e A210565 u(1, x) = 1;
%e A210565 u(2, x) = 2 + 2*x;
%e A210565 u(3, x) = 3 + 5*x + 3*x^2.
%t A210565 (* First program *)
%t A210565 u[1, x_]:= 1; v[1, x_]:= 1; z = 16;
%t A210565 u[n_, x_]:= x*u[n-1, x] + (x+1)*v[n-1, x] + 1;
%t A210565 v[n_, x_]:= x*u[n-1, x] + v[n-1, x] + 1;
%t A210565 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210565 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210565 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210565 TableForm[cu]
%t A210565 Flatten[%]    (* A210565 *)
%t A210565 Table[Expand[v[n, x]], {n, 1, z}]
%t A210565 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210565 TableForm[cv]
%t A210565 Flatten[%]    (* A210595 *)
%t A210565 (* Second program *)
%t A210565 u[n_, x_]:= u[n, x]= If[n<2, (n+1)*(1+x)^n, (1+x)*u[n-1, x] +x^2*u[n-2, x] +1+x];
%t A210565 T[n_]:= CoefficientList[Series[u[n, x], {x, 0, n}], x];
%t A210565 Table[T[n-1], {n,12}] (* _G. C. Greubel_, May 23 2021 *)
%o A210565 (Sage)
%o A210565 @CachedFunction
%o A210565 def u(n,x): return (n+1)*(1+x)^n if (n<2) else (1+x)*u(n-1,x) + x^2*u(n-2,x) +1+x
%o A210565 def T(n): return taylor( u(n,x) , x,0,n).coefficients(x, sparse=False)
%o A210565 flatten([T(n-1) for n in (1..12)]) # _G. C. Greubel_, May 23 2021
%Y A210565 Cf. A208510, A210595.
%K A210565 nonn,tabl
%O A210565 1,2
%A A210565 _Clark Kimberling_, Mar 23 2012