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A208756 Triangle of coefficients of polynomials v(n,x) jointly generated with A208755; see the Formula section. 4
1, 0, 2, 0, 1, 4, 0, 1, 3, 8, 0, 1, 3, 9, 16, 0, 1, 3, 11, 23, 32, 0, 1, 3, 13, 31, 57, 64, 0, 1, 3, 15, 39, 87, 135, 128, 0, 1, 3, 17, 47, 121, 227, 313, 256, 0, 1, 3, 19, 55, 159, 339, 579, 711, 512, 0, 1, 3, 21, 63, 201, 471, 933, 1431, 1593, 1024, 0, 1, 3, 23, 71 (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, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 02 2012

LINKS

Table of n, a(n) for n=1..71.

FORMULA

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

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

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

As triangle with 0<=k<=n : G.f.: (1-x+y*x)/(1-(1+y)*x-(2*y^2-y)*x^2). - Philippe Deléham, Mar 02 2012

T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2). - Philippe Deléham, Mar 02 2012

EXAMPLE

First five rows:

1

0...2

0...1...4

0...1...3...8

0...1...3...9...16

First five polynomials v(n,x):

1

2x

x + 4x^2

x + 3x^2 + 8x^3

x + 3x^2 + 9x^3 + 16^4

MATHEMATICA

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

u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];

v[n_, x_] := x*u[n - 1, x] + 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[%]    (* A208755 *)

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

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

TableForm[cv]

Flatten[%]    (* A208756 *)

CROSSREFS

Cf. A208755, A208510.

Sequence in context: A121298 A212206 A247489 * A259873 A121462 A271466

Adjacent sequences:  A208753 A208754 A208755 * A208757 A208758 A208759

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 01 2012

STATUS

approved

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Last modified October 15 12:31 EDT 2019. Contains 328026 sequences. (Running on oeis4.)