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A208659 Triangle of coefficients of polynomials v(n,x) jointly generated with A185045; see the Formula section. 3
1, 2, 2, 2, 6, 4, 2, 10, 16, 8, 2, 14, 36, 40, 16, 2, 18, 64, 112, 96, 32, 2, 22, 100, 240, 320, 224, 64, 2, 26, 144, 440, 800, 864, 512, 128, 2, 30, 196, 728, 1680, 2464, 2240, 1152, 256, 2, 34, 256, 1120, 3136, 5824, 7168, 5632, 2560, 512, 2, 38, 324 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Alternating row sums:  1, 0, 0, 0, 0, 0, 0, 0, 0,...

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

As triangle T(n,k) with 0<=k<=n, it is (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012

LINKS

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

FORMULA

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

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

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

As triangle T(n,k) with 0<=k<=n : T(n,k) = A029653(n,k)*2^k. - Philippe Deléham, Mar 04 2012

Sum_{k, 0<=k<=n} T(n,k)*x^k = 2*(1+x)*(1+2x)^(n-2) for n>1. - Philippe Deléham, Mar 05 2012

EXAMPLE

First five rows:

1

2...2

2...6...4

2...10...16...8

2...14...36...40...16

First five polynomials v(n,x):

1

2 + 2x = 2*(1+x)

2 + 6x + 4x^2 = 2*(1+x)*(1+2x)

2 + 10x + 16x^2 + 8x^3 = 2*(1+x)*(1+2x)^2

2 + 14x + 36x^2 + 40 x^3 + 16x^4 = 2*(1+x)*(1+2x)^3

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_] := u[n - 1, x] + 2 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[%]    (* A185045 *)

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

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

TableForm[cv]

Flatten[%]    (* A208659 *)

CROSSREFS

Cf. A185045, A208510.

Sequence in context: A054274 A053695 A210550 * A209752 A119918 A084867

Adjacent sequences:  A208656 A208657 A208658 * A208660 A208661 A208662

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 03 2012

STATUS

approved

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Last modified October 23 16:46 EDT 2019. Contains 328373 sequences. (Running on oeis4.)