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A209134 Triangle of coefficients of polynomials v(n,x) jointly generated with A209133; see the Formula section. 3
1, 0, 4, 0, 4, 10, 0, 4, 18, 28, 0, 4, 26, 72, 76, 0, 4, 34, 132, 256, 208, 0, 4, 42, 208, 572, 864, 568, 0, 4, 50, 300, 1056, 2272, 2808, 1552, 0, 4, 58, 408, 1740, 4800, 8496, 8896, 4240, 0, 4, 66, 532, 2656, 8880, 20208, 30432, 27648, 11584, 0, 4, 74 (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, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (4, -3/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 10 2012

LINKS

Table of n, a(n) for n=1..58.

FORMULA

u(n,x)=u(n-1,x)+(x+1)*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) = T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 4 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Apr 10 2012

EXAMPLE

First five rows:

1

0...4

0...4...10

0...4...18...28

0...4...26...72...76

First three polynomials v(n,x): 1, 4x, 4x + 10x^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_] := 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[%]    (* A209133 *)

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

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

TableForm[cv]

Flatten[%]    (* A209134 *)

CROSSREFS

Cf. A209133, A208510.

Sequence in context: A156732 A200341 A101980 * A281297 A058536 A154854

Adjacent sequences:  A209131 A209132 A209133 * A209135 A209136 A209137

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified October 20 20:24 EDT 2019. Contains 328273 sequences. (Running on oeis4.)