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A210561 Triangle of coefficients of polynomials u(n,x) jointly generated with A210562; see the Formula section.

%I A210561 #9 Mar 06 2017 10:46:49
%S A210561 1,1,2,1,3,4,1,3,8,8,1,3,9,20,16,1,3,9,26,48,32,1,3,9,27,72,112,64,1,
%T A210561 3,9,27,80,192,256,128,1,3,9,27,81,232,496,576,256,1,3,9,27,81,242,
%U A210561 656,1248,1280,512,1,3,9,27,81,243,716,1808,3072,2816,1024,1,3,9
%N A210561 Triangle of coefficients of polynomials u(n,x) jointly generated with A210562; see the Formula section.
%C A210561 Last term in row n:  2^(n-1)
%C A210561 Limiting row:  3^(k-1)
%C A210561 For a discussion and guide to related arrays, see A208510.
%H A210561 P. Bala, <a href="/A081577/a081577.pdf">A note on the diagonals of a proper Riordan Array</a>
%F A210561 u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
%F A210561 v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
%F A210561 where u(1,x)=1, v(1,x)=1.
%F A210561 From _Peter Bala_, Mar 06 2017: (Start)
%F A210561 T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1).
%F A210561 E.g.f. for n-th subdiagonal: exp(2*x)*(1 + x + x^2/2! + x^3/3! + ... + x^n/n!). Cf. A004070.
%F A210561 Riordan array (1/(1 - x), x*(2 + x)).
%F A210561 Row sums A048739.
%F A210561 (End)
%e A210561 First five rows:
%e A210561 1
%e A210561 1...2
%e A210561 1...3...4
%e A210561 1...3...8...8
%e A210561 1...3...9...20...16
%e A210561 First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 4x^2.
%t A210561 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210561 u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210561 v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;
%t A210561 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210561 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210561 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210561 TableForm[cu]
%t A210561 Flatten[%]  (* A210559 *)
%t A210561 Table[Expand[v[n, x]], {n, 1, z}]
%t A210561 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210561 TableForm[cv]
%t A210561 Flatten[%]  (* A210560 *)
%Y A210561 Cf. A210562, A208510, A004070, A048739.
%K A210561 nonn,tabl,easy
%O A210561 1,3
%A A210561 _Clark Kimberling_, Mar 22 2012