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Triangle of coefficients of polynomials v(n,x) jointly generated with A202390; see the Formula section.
4

%I #14 Jan 22 2020 20:13:15

%S 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 262,116,21,8,91,364,743,868,590,218,34,9,128,630,1604,2413,2240,1267,

%U 402,55,10,174,1032,3186,5926,7046,5424,2627,730,89,11,230,1617

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

%C v(n,n) = F(n+1), where F=A000045, the Fibonacci numbers.

%C Alternating row sums of v: (1,0,0,0,0,0,0,0,...).

%C 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 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 From _Philippe Deléham_, Feb 28 2012: (Start)

%F As triangle T(n,k) with 0 <= k <= n:

%F 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 G.f.: (1+y*x)/(1-2*x-y*x+x^2-y^2*x^2).

%F 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 First five rows:

%e 1;

%e 2, 2;

%e 3, 6, 3;

%e 4, 13, 14, 5;

%e 5, 24, 41, 30, 8;

%e The first five polynomials v(n,x):

%e 1

%e 2 + 2x

%e 3 + 6x + 3x^2

%e 4 + 13x + 14x^2 + 5x^3

%e 5 + 24x + 41x^2 + 30x^3 + 8x^4

%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 + 1)*u[n - 1, x] + (x + 1)*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[%] (* A202390 *)

%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[%] (* A208340 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* u row sums *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* v row sums *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* u alt. row sums *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* v alt. row sums *)

%Y Cf. A202390.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Feb 27 2012