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A207613 Triangle of coefficients of polynomials v(n,x) jointly generated with A207612; see Formula section.

%I A207613 #10 Jan 02 2016 14:04:50
%S A207613 1,2,2,3,4,4,5,8,8,8,8,16,20,16,16,13,30,44,48,32,32,21,56,92,112,112,
%T A207613 64,64,34,102,188,256,272,256,128,128,55,184,372,560,672,640,576,256,
%U A207613 256,89,328,724,1184,1552,1696,1472,1280,512,512,144,580,1384
%N A207613 Triangle of coefficients of polynomials v(n,x) jointly generated with A207612; see Formula section.
%C A207613 Only column 1 contains odd numbers.
%C A207613 column 1:  A000045 (Fibonacci sequence)
%C A207613 row sums:  A002878 (bisection of Lucas sequence)
%C A207613 top edge:  A000079 (powers of 2)
%F A207613 u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = u(n-1,x) + 2x*v(n-1,x) + 1, where u(1,x) = 1, v(1,x) = 1.
%F A207613 With offset 0, the Riordan array ((1 + z)/(1 - z - z^2), 2*z*(1 - z)/(1 - z - z^2)) with o.g.f. (1 + z)/(1 - z - z^2 - x*(2*z - 2*z^2)) = 1 + (2 + 2*x)*z + (3 + 4*x + 4*x^2)*z^2 + .... - _Peter Bala_, Dec 30 2015
%e A207613 First five rows:
%e A207613   1
%e A207613   2  2
%e A207613   3  4  4
%e A207613   5  8  8  8
%e A207613   8 16 20 16 16
%t A207613 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207613 u[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207613 v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x] + 1
%t A207613 Table[Factor[u[n, x]], {n, 1, z}]
%t A207613 Table[Factor[v[n, x]], {n, 1, z}]
%t A207613 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207613 TableForm[cu]
%t A207613 Flatten[%]    (* A207612 *)
%t A207613 Table[Expand[v[n, x]], {n, 1, z}]
%t A207613 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207613 TableForm[cv]
%t A207613 Flatten[%]    (* A207613 *)
%Y A207613 A000045 (column 1), A000079 (main diagonal), A002878 (row sums). Cf. A207612, A208510.
%K A207613 nonn,tabl,easy
%O A207613 1,2
%A A207613 _Clark Kimberling_, Feb 19 2012