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A209129 Triangle of coefficients of polynomials v(n,x) jointly generated with A209128; see the Formula section. 3
1, 0, 3, 0, 2, 7, 0, 2, 8, 17, 0, 2, 10, 28, 41, 0, 2, 12, 42, 88, 99, 0, 2, 14, 58, 154, 262, 239, 0, 2, 16, 76, 240, 524, 752, 577, 0, 2, 18, 96, 348, 908, 1692, 2104, 1393, 0, 2, 20, 118, 480, 1440, 3224, 5260, 5776, 3363, 0, 2, 22, 142, 638, 2148, 5544 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

As triangle T(n,k) with 0<=k<=n, it is (0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

LINKS

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

FORMULA

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

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

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

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 3 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 21 2012

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

EXAMPLE

First five rows:

1

0...3

0...2....7

0...2....8...17

0...2...10...28...41

First three polynomials v(n,x): 1, 3x, 2x + 7x^2.

MATHEMATICA

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

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

v[n_, x_] := x*u[n - 1, x] + 2 x*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[%]    (* A209128 *)

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

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

TableForm[cv]

Flatten[%]    (* A209129 *)

CROSSREFS

Cf. A209128, A208510.

Sequence in context: A298707 A274417 A208764 * A282694 A011075 A248820

Adjacent sequences:  A209126 A209127 A209128 * A209130 A209131 A209132

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified October 15 23:54 EDT 2019. Contains 328038 sequences. (Running on oeis4.)