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A210596 Triangle read by rows of coefficients of polynomials v(n,x) jointly generated with A210221; see the Formula section.

%I A210596 #38 Jun 27 2025 23:25:27
%S A210596 1,1,2,2,2,4,3,6,4,8,5,10,16,8,16,8,20,28,40,16,32,13,36,64,72,96,32,
%T A210596 64,21,66,124,184,176,224,64,128,34,118,248,376,496,416,512,128,256,
%U A210596 55,210,476,808,1056,1280,960,1152,256,512,89,370,908,1640,2416
%N A210596 Triangle read by rows of coefficients of polynomials v(n,x) jointly generated with A210221; see the Formula section.
%C A210596 Row n begins with F(n) and ends with 2^(n-1), where F = A000045 (Fibonacci numbers)
%C A210596 Row sums:  odd-indexed Fibonacci numbers, see A001519.
%C A210596 For a discussion and guide to related arrays, see A208510.
%C A210596 Riordan array (1/(1 - z - z^2), 2*z*(1 - z)/(1 - z - z^2)). - _Peter Bala_, Dec 30 2015
%H A210596 G. C. Greubel, <a href="/A210596/b210596.txt">Rows n=1..102 of triangle, flattened</a>
%F A210596 u(n,x) = u(n-1,x) + v(n-1,x),
%F A210596 v(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A210596 where u(1,x) = 1, v(1,x) = 1.
%F A210596 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-1), T(1,0) = T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 25 2012
%F A210596 G.f.: 1/((1-x-x^2) - t*2*x*(1-x)). - _G. C. Greubel_, Dec 15 2018
%e A210596 First five rows:
%e A210596   1
%e A210596   1  2
%e A210596   2  2  4
%e A210596   3  6  4  8
%e A210596   5 10 16  8 16
%e A210596 First three polynomials v(n,x): 1, 1 + 2x, 2 + 2x + 4x^2.
%t A210596 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210596 u[n_, x_] := u[n - 1, x] + v[n - 1, x];
%t A210596 v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A210596 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210596 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210596 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210596 TableForm[cu]
%t A210596 Flatten[%]   (* A210221 *)
%t A210596 Table[Expand[v[n, x]], {n, 1, z}]
%t A210596 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210596 TableForm[cv]
%t A210596 Flatten[%]   (* A210596 *)
%t A210596 With[{m = 10}, CoefficientList[CoefficientList[Series[1/((1-x-x^2) - t*2*x*(1-x)), {x, 0, m}, {t, 0, m}], x], t]]//Flatten (* _G. C. Greubel_, Dec 15 2018 *)
%o A210596 (Python)
%o A210596 from sympy import Poly
%o A210596 from sympy.abc import x
%o A210596 def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
%o A210596 def v(n, x): return 1 if n==1 else u(n - 1, x) + 2*x*v(n - 1, x)
%o A210596 def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
%o A210596 for n in range(1, 13): print (a(n)) # _Indranil Ghosh_, May 27 2017
%o A210596 (PARI) {T(n,k) = if(n==1 && k==0, 1, if(n==2 && k==0, 1, if(n==2 && k==1, 2, if(k<0 || k>n-1, 0, T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-1) ))))};
%o A210596 for(n=1,15, for(k=0, n-1, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Dec 15 2018
%Y A210596 Cf. A000045, A000079, A001519, A210221, A208510.
%K A210596 nonn,tabl
%O A210596 1,3
%A A210596 _Clark Kimberling_, Mar 24 2012