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A209687 Triangle of coefficients of polynomials u(n,x) jointly generated with A208339; see the Formula section. 2
1, 0, 2, 0, 1, 5, 0, 1, 5, 12, 0, 1, 6, 18, 29, 0, 1, 7, 26, 58, 70, 0, 1, 8, 35, 98, 175, 169, 0, 1, 9, 45, 149, 339, 507, 408, 0, 1, 10, 56, 212, 574, 1108, 1428, 985, 0, 1, 11, 68, 288, 894, 2066, 3476, 3940, 2378, 0, 1, 12, 81, 378, 1314, 3492, 7074, 10572 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Alternating row sums: signed powers of 2.  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 (2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 28 2012

LINKS

Table of n, a(n) for n=1..64.

FORMULA

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

v(n,x)=2x*u(n-1,x)+(x+1)*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) = 2 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 28 2012

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

EXAMPLE

First five rows:

1

0...2

0...1...5

0...1...5...12

0...1...6...18...29

First three polynomials v(n,x): 1, 2x, x + 5x^2.

MATHEMATICA

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

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

v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*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[%]   (* A209687 *)

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

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

TableForm[cv]

Flatten[%]   (* A208339 *)

CROSSREFS

Cf. A208339, A208510.

Sequence in context: A293961 A112899 A212808 * A108263 A244523 A134433

Adjacent sequences:  A209684 A209685 A209686 * A209688 A209689 A209690

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 12 2012

STATUS

approved

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Last modified July 21 17:54 EDT 2019. Contains 325198 sequences. (Running on oeis4.)