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

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, 262, 116, 21, 8, 91, 364, 743, 868, 590, 218, 34, 9, 128, 630, 1604, 2413, 2240, 1267, 402, 55, 10, 174, 1032, 3186, 5926, 7046, 5424, 2627, 730, 89, 11, 230, 1617

Offset

1,2

Comments

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

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

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.

Links

Table of n, a(n) for n=1..58.

Formula

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.

From Philippe Deléham, Feb 28 2012: (Start)

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

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.

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

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)

Example

First five rows:
  1;
  2,  2;
  3,  6,  3;
  4, 13, 14,  5;
  5, 24, 41, 30,  8;
The first five polynomials v(n,x):
  1
  2 +  2x
  3 +  6x +  3x^2
  4 + 13x + 14x^2 +  5x^3
  5 + 24x + 41x^2 + 30x^3 + 8x^4

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 13;
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];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A202390 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208340 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* u row sums *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* v row sums *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* u alt. row sums *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* v alt. row sums *)

Crossrefs

Cf. A202390.

Sequence in context: A075196 A196912 A197079 * A308503 A196967 A210859

Adjacent sequences: A208337 A208338 A208339 * A208341 A208342 A208343

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 27 2012

Status

approved