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Triangle of coefficients of polynomials u(n,x) jointly generated with A210595; see the Formula section.
3

%I #12 May 25 2021 01:37:47

%S 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 71,21,8,35,98,190,256,233,130,34,9,44,140,315,511,594,474,235,55,10,

%U 54,192,490,924,1295,1324,942,420,89,11,65,255,726,1554,2534,3130,2860,1836,744,144

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

%C Row n starts with n and ends with F(n+1), where F=A000045 (Fibonacci numbers).

%C Row sums: A005409.

%C Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0, ...

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

%H G. C. Greubel, <a href="/A210565/b210565.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]( 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 First five rows:

%e 1;

%e 2, 2;

%e 3, 5, 3;

%e 4, 9, 10, 5;

%e 5, 14, 22, 20, 8;

%e First three polynomials u(n,x):

%e u(1, x) = 1;

%e u(2, x) = 2 + 2*x;

%e u(3, x) = 3 + 5*x + 3*x^2.

%t (* First program *)

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

%t u[n_, x_]:= x*u[n-1, x] + (x+1)*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 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 T[n_]:= CoefficientList[Series[u[n, x], {x, 0, n}], x];

%t Table[T[n-1], {n,12}] (* _G. C. Greubel_, May 23 2021 *)

%o (Sage)

%o @CachedFunction

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

%o flatten([T(n-1) for n in (1..12)]) # _G. C. Greubel_, May 23 2021

%Y Cf. A208510, A210595.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 23 2012