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A210791 Triangle of coefficients of polynomials u(n,x) jointly generated with A210792; see the Formula section. 3
1, 1, 1, 1, 2, 2, 1, 3, 7, 3, 1, 4, 17, 14, 5, 1, 5, 36, 42, 30, 8, 1, 6, 72, 104, 111, 58, 13, 1, 7, 141, 233, 329, 251, 111, 21, 1, 8, 275, 494, 862, 848, 553, 206, 34, 1, 9, 538, 1016, 2097, 2479, 2112, 1158, 377, 55, 1, 10, 1058, 2056, 4870, 6608, 6875 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers).

Column 2: 1,2,3,4,5,6,7,8,...

Row sums: A007051.

Alternating row sums: A000129.

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

Subtriangle of the triangle given by (1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 29 2012

LINKS

Table of n, a(n) for n=1..62.

FORMULA

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

v(n,x) = (x-1)*u(n-1,x) + (x+2)*v(n-1,x),

where u(1,x)=1, v(1,x)=1.

From Philippe Deléham, Mar 29 2012: (Start)

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

G.f.: (1 - 2*x - y*x + 2*y*x^2 - y^2*x^2)/(1 - 3*x - y*x + 2*x^2 + 2*y*x^2 - y^2*x^2).

T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - 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)

EXAMPLE

First five rows:

  1;

  1,  1;

  1,  2,  2;

  1,  3,  7,  3;

  1,  4, 17, 14,  5;

First three polynomials u(n,x):

  1

  1 + x

  1 + 2x + 2x^2.

From Philippe Deléham, Mar 29 2012: (Start)

(1, 0, 0, 2, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins:

  1;

  1,   0;

  1,   1,   0;

  1,   2,   2,   0;

  1,   3,   7,   3,   0;

  1,   4,  17,  14,   5,   0;

  1,   5,  36,  42,  30,   8,   0;

  1,   6,  72, 104, 111,  58,  13,   0; (End)

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 = 2; 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[%]    (* A210791 *)

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

TableForm[cv]

Flatten[%]    (* A210792 *)

Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A007051 *)

Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000244 *)

Table[u[n, x] /. x -> -1, {n, 1, z}] (* A001129 *)

Table[v[n, x] /. x -> -1, {n, 1, z}] (* A001333 *)

CROSSREFS

Cf. A210792, A208510.

Sequence in context: A158497 A334894 A110564 * A299500 A330141 A007441

Adjacent sequences:  A210788 A210789 A210790 * A210792 A210793 A210794

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 26 2012

STATUS

approved

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Last modified September 28 13:24 EDT 2020. Contains 337393 sequences. (Running on oeis4.)