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A208613 Triangle of coefficients of polynomials v(n,x) jointly generated with A208612; see the Formula section.

%I A208613 #6 Mar 30 2012 18:58:13
%S A208613 1,3,2,5,7,3,7,17,16,5,9,34,51,33,8,11,60,127,129,65,13,13,97,272,386,
%T A208613 302,124,21,15,147,525,975,1052,666,231,34,17,212,938,2186,3049,2646,
%U A208613 1409,423,55,19,294,1578,4482,7757,8650,6285,2887,764,89,21
%N A208613 Triangle of coefficients of polynomials v(n,x) jointly generated with A208612; see the Formula section.
%C A208613 Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,...
%F A208613 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208613 v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A208613 where u(1,x)=1, v(1,x)=1.
%e A208613 First five rows:
%e A208613 1
%e A208613 3...2
%e A208613 5...7....3
%e A208613 7...17...16...3
%e A208613 9...34...51...33...8
%e A208613 First five polynomials v(n,x):
%e A208613 1
%e A208613 3 + 2x
%e A208613 5 + 7x + 3x^2
%e A208613 7 + 17x + 16x^2 + 3x^3
%e A208613 9 + 34x + 51x^2 + 33x^3 + 8x^4
%t A208613 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208613 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208613 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A208613 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208613 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208613 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208613 TableForm[cu]
%t A208613 Flatten[%]  (* A208612 *)
%t A208613 Table[Expand[v[n, x]], {n, 1, z}]
%t A208613 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208613 TableForm[cv]
%t A208613 Flatten[%]  (* A208613 *)
%Y A208613 Cf. A208612.
%K A208613 nonn,tabl
%O A208613 1,2
%A A208613 _Clark Kimberling_, Mar 01 2012