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A208340 Triangle of coefficients of polynomials v(n,x) jointly generated with A202390; see the Formula section.

%I A208340 #14 Jan 22 2020 20:13:15
%S A208340 1,2,2,3,6,3,4,13,14,5,5,24,41,30,8,6,40,96,109,60,13,7,62,196,308,
%T A208340 262,116,21,8,91,364,743,868,590,218,34,9,128,630,1604,2413,2240,1267,
%U A208340 402,55,10,174,1032,3186,5926,7046,5424,2627,730,89,11,230,1617
%N A208340 Triangle of coefficients of polynomials v(n,x) jointly generated with A202390; see the Formula section.
%C A208340 v(n,n) = F(n+1), where F=A000045, the Fibonacci numbers.
%C A208340 Alternating row sums of v: (1,0,0,0,0,0,0,0,...).
%C A208340 As triangle T(n,k) with 0 <= k <= n, it is (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%F A208340 u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x), where u(1,x)=1, v(1,x)=1.
%F A208340 From _Philippe Deléham_, Feb 28 2012: (Start)
%F A208340 As triangle T(n,k) with 0 <= k <= n:
%F A208340 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-2) with T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k < 0 or if k > n.
%F A208340 G.f.: (1+y*x)/(1-2*x-y*x+x^2-y^2*x^2).
%F A208340 Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000027(n+1), A003946(n), A109115(n), A180031(n) for x = -1, 0, 1, 2, 3 respectively. (End)
%e A208340 First five rows:
%e A208340   1;
%e A208340   2,  2;
%e A208340   3,  6,  3;
%e A208340   4, 13, 14,  5;
%e A208340   5, 24, 41, 30,  8;
%e A208340 The first five polynomials v(n,x):
%e A208340   1
%e A208340   2 +  2x
%e A208340   3 +  6x +  3x^2
%e A208340   4 + 13x + 14x^2 +  5x^3
%e A208340   5 + 24x + 41x^2 + 30x^3 + 8x^4
%t A208340 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208340 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208340 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A208340 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208340 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208340 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208340 TableForm[cu]
%t A208340 Flatten[%]    (* A202390 *)
%t A208340 Table[Expand[v[n, x]], {n, 1, z}]
%t A208340 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208340 TableForm[cv]
%t A208340 Flatten[%]    (* A208340 *)
%t A208340 Table[u[n, x] /. x -> 1, {n, 1, z}]  (* u row sums *)
%t A208340 Table[v[n, x] /. x -> 1, {n, 1, z}]  (* v row sums *)
%t A208340 Table[u[n, x] /. x -> -1, {n, 1, z}] (* u alt. row sums *)
%t A208340 Table[v[n, x] /. x -> -1, {n, 1, z}] (* v alt. row sums *)
%Y A208340 Cf. A202390.
%K A208340 nonn,tabl
%O A208340 1,2
%A A208340 _Clark Kimberling_, Feb 27 2012