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A209171 Triangle of coefficients of polynomials v(n,x) jointly generated with A209170; see the Formula section. 3
1, 3, 2, 6, 8, 3, 12, 25, 19, 5, 24, 68, 77, 40, 8, 48, 172, 259, 201, 80, 13, 96, 416, 782, 806, 478, 154, 21, 192, 976, 2200, 2825, 2222, 1067, 289, 34, 384, 2240, 5888, 9048, 8857, 5640, 2277, 532, 55, 768, 5056, 15184, 27160, 31787, 25184, 13483 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Column 1:  Fibonacci numbers (A000045).

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

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

Subtriangle of (1, 2, -3/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 10 2012

LINKS

Table of n, a(n) for n=1..52.

FORMULA

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

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

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

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 3, T(2,1) = 2 . - Philippe Deléham, Mar 10 2012

Sum_{k, T(n,k)*x^k = A000012(n), A003945(n-1), A007483(n-1) for x = -1, 0, 1 respectively. - Philippe Deléham, Mar 10 2012

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

EXAMPLE

First five rows:

1

3...2

6...8...3

12...25...19...5

24...68...77...40...8

First three polynomials v(n,x): 1, 3 + 2x, 6 + 8x + 3x^2.

Triangle (1, 2, -3/2, 1/2, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, ...) begins (0<=k<=n) :

1

1, 0

3, 2, 0

6, 8, 3, 0

12, 25, 19, 5, 0

24, 68, 77, 40, 8, 0

48, 172, 259, 201, 80, 13, 0

96, 416, 782, 806, 478, 154, 21, 0

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_] := (x + 1)*u[n - 1, x] + (x + 1)*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[%]    (* A209170 *)

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

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

TableForm[cv]

Flatten[%]    (* A209171 *)

CROSSREFS

Cf. A209170, A208510.

Sequence in context: A127717 A210236 A193998 * A160855 A120232 A292961

Adjacent sequences:  A209168 A209169 A209170 * A209172 A209173 A209174

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 08 2012

STATUS

approved

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Last modified October 20 08:05 EDT 2019. Contains 328252 sequences. (Running on oeis4.)