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A208748 Triangle of coefficients of polynomials v(n,x) jointly generated with A208747; see the Formula section. 4
1, 0, 4, 0, 2, 12, 0, 2, 8, 40, 0, 2, 8, 40, 128, 0, 2, 8, 48, 160, 416, 0, 2, 8, 56, 208, 640, 1344, 0, 2, 8, 64, 256, 928, 2432, 4352, 0, 2, 8, 72, 304, 1248, 3840, 9088, 14080, 0, 2, 8, 80, 352, 1600, 5504, 15616, 33280, 45568, 0, 2, 8, 88, 400, 1984, 7424 (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, ...) DELTA (4, -1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 14 2012

LINKS

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

FORMULA

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

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

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

T(n,k) = 2^k*A208343(n,k). - Philippe Deléham, Mar 05 2012

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 4*T(n-2,k-2), T(1,0) =   1, T(2,0) = T(3,0) = 0, T(2,1) = 4, T(3,1) = 2, T(3,2) = 12, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 14 2012

G.f.: (-1+x-2*x*y)*x*y/(-1+x+2*x*y-2*x^2*y+4*x^2*y^2). - R. J. Mathar, Aug 11 2015

EXAMPLE

First five rows:

1

0...4

0...2...12

0...2...8...40

0...2...8...40...128

First five polynomials v(n,x):

1

4x

2x + 12x^2

2x + 8x^2 + 40x^3

2x + 8x^2 + 40x^3 + 128x^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_] := 2 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[%]    (* A208747 *)

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

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

TableForm[cv]

Flatten[%]    (* A208748 *)

CROSSREFS

Cf. A208747, A208510.

Sequence in context: A282279 A208333 A279413 * A134895 A318468 A145877

Adjacent sequences:  A208745 A208746 A208747 * A208749 A208750 A208751

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 02 2012

STATUS

approved

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Last modified October 21 16:25 EDT 2019. Contains 328302 sequences. (Running on oeis4.)