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%I #24 Jun 06 2024 08:23:51
%S 1,1,1,1,1,3,1,1,4,7,1,1,5,10,17,1,1,6,13,27,41,1,1,7,16,38,71,99,1,1,
%T 8,19,50,106,186,239,1,1,9,22,63,146,294,484,577,1,1,10,25,77,191,424,
%U 806,1253,1393,1,1,11,28,92,241,577,1212,2191,3229,3363,1,1,12
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208345; see the Formula section.
%C Row sums, u(n,1): (1,2,5,13,...), odd-indexed Fibonacci numbers.
%C Row sums, v(n,1): (1,3,8,21,...), even-indexed Fibonacci numbers.
%C Subtriangle of the triangle given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 09 2012
%F u(n,x) = u(n-1,x) + x*v(n-1,x),
%F v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F From _Philippe Deléham_, Apr 09 2012: (Start)
%F As DELTA-triangle T(n,k) with 0 <= k <= n:
%F G.f.: (1-2*y*x+y*x^2-y^2*x^2)/(1-x-2*y*x+2*y*x^2-y^2*x^2).
%F T(n,k) = T(n-1,k) + 2*T(n-1,k-1) -2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%F Working with an offset of 0, the row reversed triangle is the Riordan array ( (1 - x)/(1 - 2*x - x^2), x*(1 - 2*x)/(1 - 2*x - x^2) ) with g.f. (1 - x)/(1 - (2 + y)*x - (1 - 2*y)*x^2) = 1 + (1 + y)*x + (3 + y + y^2)*x^2 + (7 + 4*y + y^2 + y^3)*x^3 + .... - _Peter Bala_, Jun 01 2024
%e First five rows:
%e 1;
%e 1, 1;
%e 1, 1, 3;
%e 1, 1, 4, 7;
%e 1, 1, 5, 10, 17;
%e First five polynomials u(n,x):
%e 1
%e 1 + x
%e 1 + x + 3x^2
%e 1 + x + 4x^2 + 7x^3
%e 1 + x + 5x^2 + 10x^3 + 17x^4.
%e From _Philippe Deléham_, Apr 09 2012: (Start)
%e (1, 0, -1, 1, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, ...) begins:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 1, 3, 0;
%e 1, 1, 4, 7, 0;
%e 1, 1, 5, 10, 17, 0;
%e 1, 1, 6, 13, 27, 41, 0; (End)
%t u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t v[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x];
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A208344 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A208345 *)
%t Table[u[n, x] /. x -> 1, {n, 1, z}]
%t Table[v[n, x] /. x -> 1, {n, 1, z}]
%Y Cf. A208345, A208510.
%K nonn,tabl
%O 1,6
%A _Clark Kimberling_, Feb 25 2012
%E a(69) corrected by _Georg Fischer_, Sep 03 2021
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