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A210042 Triangle of coefficients of polynomials u(n,x) jointly generated with A124927; see the Formula section. 3
1, 3, 5, 2, 7, 6, 2, 9, 12, 8, 2, 11, 20, 20, 10, 2, 13, 30, 40, 30, 12, 2, 15, 42, 70, 70, 42, 14, 2, 17, 56, 112, 140, 112, 56, 16, 2, 19, 72, 168, 252, 252, 168, 72, 18, 2, 21, 90, 240, 420, 504, 420, 240, 90, 20, 2, 23, 110, 330, 660, 924, 924, 660, 330, 110 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Row sums:  A000225

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

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

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

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

Subtriangle of the triangle given by (1, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.- Philippe Deléham, Mar 25 2012

LINKS

Table of n, a(n) for n=1..65.

FORMULA

First five rows:

1

3

5...2

7...6....2

9...12...8...2

First three polynomials u(n,x): 1, 3, 5 + 2x.

Also, counting the top row as row 0, row n for n>0 is as

follows: 2n+1, 2C(n,2), 2C(n,3),..., 2C(n,n).

Contribution from Philippe Deléham, Mar 25 2012. (Start)

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

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

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0) = T(1,0) = 1, T(2,0) = 3, T(1,1) = T(2,1) = T(2,2) = 0 and T(n,k) = 0 if k<0 or if k>n. (End)

g.f.: (1+x-x*y)*x*y/((-1+x)*(x+x*y-1)). - R. J. Mathar, Aug 12 2015

EXAMPLE

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

1

1, 0

3, 0, 0

5, 2, 0, 0

7, 6, 2, 0, 0

9, 12, 8, 2, 0, 0

11, 20, 20, 10, 2, 0, 0

13, 30, 40, 30, 12, 2, 0, 0. - Philippe Deléham, Mar 25 2012

MATHEMATICA

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

u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;

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

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[%]    (* A210042 *)

Table[Expand[v[n, x]], {n, 1, z}]

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

TableForm[cv]

Flatten[%]    (* A124927 *)

Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)

Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)

Table[u[n, x] /. x -> -1, {n, 1, z}] (* A010701 *)

Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000012 signed *)

CROSSREFS

Cf. A124927, A208510.

Sequence in context: A285297 A097465 A120683 * A274421 A079313 A125132

Adjacent sequences:  A210039 A210040 A210041 * A210043 A210044 A210045

KEYWORD

nonn,tabf

AUTHOR

Clark Kimberling, Mar 17 2012

STATUS

approved

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Last modified October 15 10:46 EDT 2019. Contains 328026 sequences. (Running on oeis4.)