login
Triangle of coefficients of polynomials v(n,x) jointly generated with A210561; see the Formula section.
3

%I #13 Mar 06 2017 10:46:59

%S 1,2,2,2,5,4,2,6,12,8,2,6,17,28,16,2,6,18,46,64,32,2,6,18,53,120,144,

%T 64,2,6,18,54,152,304,320,128,2,6,18,54,161,424,752,704,256,2,6,18,54,

%U 162,474,1152,1824,1536,512,2,6,18,54,162,485,1372,3056,4352

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210561; see the Formula section.

%C Last term in row n: 2^(n-1)

%C Limiting row: 2*3^(n-1)

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

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

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

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

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

%C First five rows:

%C 1

%C 2...2

%C 2...5...4

%C 2...6...12...8

%C 2...6...17...28...16

%C First three polynomials v(n,x): 1, 2 + 2x , 2 + 5x + 4x^2.

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

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

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

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

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

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

%C TableForm[cu]

%C Flatten[%] (* A210561 *)

%C Table[Expand[v[n, x]], {n, 1, z}]

%C cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%C TableForm[cv]

%C Flatten[%] (* A210562 *)

%C Cf. A210561, A208510.

%H P. Bala, <a href="/A081577/a081577.pdf">A note on the diagonals of a proper Riordan Array</a>

%F From _Peter Bala_, Mar 06 2017: (Start)

%F T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1).

%F E.g.f for the n-th subdiagonal: exp(2*x)*(2 + 2*x + 2*x^2/2! + 2*x^3/3! + ... + 2*x^(n-1)/(n-1)! + x^n/n!).

%F Riordan array ((1 + x)/(1 - x), x*(2 + x)).

%F Row sums A005409 (except for the initial term).

%F (End)

%Y Row sums A005409. Cf. A208510, A210561.

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Mar 22 2012