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

%I A210562 #17 Jul 30 2025 00:56:54
%S A210562 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 A210562 64,2,6,18,54,152,304,320,128,2,6,18,54,161,424,752,704,256,2,6,18,54,
%U A210562 162,474,1152,1824,1536,512,2,6,18,54,162,485,1372,3056,4352
%N A210562 Triangle of coefficients of polynomials v(n,x) jointly generated with A210561; see the Formula section.
%C A210562 Last term in row n:  2^(n-1)
%C A210562 Limiting row:  2*3^(n-1)
%C A210562 Alternating row sums: 1,0,1,0,1,0,1,0,...
%C A210562 For a discussion and guide to related arrays, see A208510.
%H A210562 P. Bala, <a href="/A081577/a081577.pdf">A note on the diagonals of a proper Riordan Array</a>
%F A210562 u(n,x) = x*u(n-1,x)+x*v(n-1,x)+1,
%F A210562 v(n,x) = (x+1)*u(n-1,x)+x*v(n-1,x)+1,
%F A210562 where u(1,x)=1, v(1,x)=1.
%F A210562 From _Peter Bala_, Mar 06 2017: (Start)
%F A210562 T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1).
%F A210562 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 A210562 Riordan array ((1 + x)/(1 - x), x*(2 + x)).
%F A210562 Row sums A005409 (except for the initial term). (End)
%e A210562 First five rows:
%e A210562   1
%e A210562   2   2
%e A210562   2   5   4
%e A210562   2   6   12   8
%e A210562   2   6   17   28   16
%e A210562 First three polynomials v(n,x): 1, 2 + 2*x, 2 + 5*x + 4*x^2.
%t A210562 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210562 u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210562 v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210562 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210562 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210562 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210562 TableForm[cu]
%t A210562 Flatten[%]  (* A210561 *)
%t A210562 Table[Expand[v[n, x]], {n, 1, z}]
%t A210562 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210562 TableForm[cv]
%t A210562 Flatten[%]  (* A210562 *)
%Y A210562 Row sums A005409. Cf. A208510, A210561.
%K A210562 nonn,tabl,easy
%O A210562 1,2
%A A210562 _Clark Kimberling_, Mar 22 2012