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A210804 Triangle of coefficients of polynomials v(n,x) jointly generated with A210803; see the Formula section. 3
1, 2, 2, 5, 8, 3, 14, 27, 18, 5, 41, 88, 79, 40, 8, 122, 284, 310, 215, 80, 13, 365, 912, 1152, 980, 510, 156, 21, 1094, 2917, 4144, 4091, 2660, 1150, 294, 34, 3281, 9296, 14578, 16176, 12393, 6752, 2461, 544, 55, 9842, 29526, 50436, 61638, 53730 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row n ends with F(n), where F=A000045 (Fibonacci numbers).

Column 1:  A007051.

Row sums:  A000302 (powers of 4).

Alternating row sums:  1,0,0,0,0,0,0,0,0,...

For a discussion and guide to related arrays, see A208510.

Essentially the same triangle as given by (2, 1/2, 3/2, 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. - Philippe Deléham Jul 11 2012

LINKS

Table of n, a(n) for n=1..50.

FORMULA

u(n,x) = u(n-1,x) + x*v(n-1,x) + 1, v(n,x) = (x-1)*u(n-1,x) + (x+3)*v(n-1,x), where u(1,x)=1, v(1,x)=1.

T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = T(2,1) = 2, T(3,0) = 5, T(3,1) = 8, T(3,2) = 3, T(n,k) = 0 if k < 0 or if k >= n. - Philippe Deléham, Jul 11 2012

G.f.: (-1+2*x-x*y)*x*y/(-1+4*x+x*y-3*x^2-2*x^2*y+x^2*y^2). - R. J. Mathar, Aug 12 2015

EXAMPLE

First five rows:

   1;

   2,  2;

   5,  8,  3;

  14, 27, 18,  5;

  41, 88, 79, 40,  8;

First three polynomials v(n,x):

  1

  2 + 2x

  5 + 8x + 3x^2

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 16;

u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;

d[x_] := h + x; e[x_] := p + x;

v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;

j = 0; c = 0; h = -1; p = 3; f = 0;

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[%]    (* A210803 *)

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]    (* A210804 *)

Table[u[n, x] /. x -> 1, {n, 1, z}]   (* A047849 *)

Table[v[n, x] /. x -> 1, {n, 1, z}]   (* A000302 *)

Table[u[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)

Table[v[n, x] /. x -> -1, {n, 1, z}]  (* A000007 *)

CROSSREFS

Cf. A210803, A208510.

Sequence in context: A210637 A201972 A202396 * A087910 A327597 A284325

Adjacent sequences:  A210801 A210802 A210803 * A210805 A210806 A210807

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 27 2012

STATUS

approved

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Last modified November 27 23:47 EST 2020. Contains 338685 sequences. (Running on oeis4.)