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Triangle of coefficients of polynomials v(n,x) jointly generated with A209999; see the Formula section.
3

%I #10 May 25 2021 01:37:56

%S 1,2,1,3,3,2,4,6,7,3,5,10,16,13,5,6,15,30,35,25,8,7,21,50,75,76,46,13,

%T 8,28,77,140,181,157,84,21,9,36,112,238,371,413,317,151,34,10,45,156,

%U 378,686,924,911,625,269,55,11,55,210,570,1176,1848,2206,1949,1211,475,89

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A209999; see the Formula section.

%C Row n starts with n and ends with F(n), where F=A000045 (Fibonacci numbers).

%C Row sums: A048739.

%C Alternating row sums: 1,1,2,2,3,3,4,4,5,5, ...

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

%H G. C. Greubel, <a href="/A210595/b210595.txt">Rows n = 1..30 of the triangle, flattened</a>

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

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

%F where u(1,x) = 1, v(1,x) = 1.

%F T(n, k) = [x^k]( v(n,x) ), where v(n, x) = (1+x)*v(n-1, x) + x^2*v(n-2, x) + 1, v(1, x) = 1, and v(2, x) = 2 + x. - _G. C. Greubel_, May 24 2021

%e First few rows are:

%e 1;

%e 2, 1;

%e 3, 3, 2;

%e 4, 6, 7, 3;

%e 5, 10, 16, 13, 5;

%e 6, 15, 30, 35, 25, 8;

%e 7, 21, 50, 75, 76, 46, 13;

%e First few polynomials v(n,x) are:

%e v(1, x) = 1;

%e v(2, x) = 2 + 1*x;

%e v(3, x) = 3 + 3*x + 2*x^2;

%e v(4, x) = 4 + 6*x + 7*x^2 + 3*x^3;

%e v(5, x) = 5 + 10*x + 16*x^2 + 13*x^3 + 5*x^4;

%t (* First program *)

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

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

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

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

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

%t TableForm[cu]

%t Flatten[%] (* A210565 *)

%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[%] (* A210595 *)

%t (* Second program *)

%t v[n_, x_]:= v[n, x]= If[n<2, n+1 +n*x, (1+x)*v[n-1, x] +x^2*v[n-2, x] +1];

%t T[n_]:= CoefficientList[Series[v[n, x], {x,0,n}], x];

%t Table[T[n-1], {n, 12}]//Flatten (* _G. C. Greubel_, May 24 2021 *)

%o (Sage)

%o @CachedFunction

%o def v(n,x): return n+1+n*x if (n<2) else (1+x)*v(n-1,x) +x^2*v(n-2,x) +1

%o def T(n): return taylor( v(n,x) , x,0,n).coefficients(x, sparse=False)

%o flatten([T(n-1) for n in (1..12)]) # _G. C. Greubel_, May 24 2021

%Y Cf. A208510, A210565.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 23 2012