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A208343 Triangle of coefficients of polynomials v(n,x) jointly generated with A208342; see the Formula section. 5
1, 0, 2, 0, 1, 3, 0, 1, 2, 5, 0, 1, 2, 5, 8, 0, 1, 2, 6, 10, 13, 0, 1, 2, 7, 13, 20, 21, 0, 1, 2, 8, 16, 29, 38, 34, 0, 1, 2, 9, 19, 39, 60, 71, 55, 0, 1, 2, 10, 22, 50, 86, 122, 130, 89, 0, 1, 2, 11, 25, 62, 116, 187, 241, 235, 144, 0, 1, 2, 12, 28, 75, 150, 267, 392, 468 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

u(n,n) = A000045(n+1) (Fibonacci numbers).

n-th row sum:  2^(n-1)

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, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

LINKS

Table of n, a(n) for n=1..76.

FORMULA

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

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

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

(Start) As triangle T(n,k) with 0<=k<=n :

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k>n or if k<0.

G.f.: (1-(1-y)*x)/(1-(1+y)*x+y*(1-y)*x^2).

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)*A091003(n+1), A152166(n), A000007(n), A000079(n), A055099(n), A152224(n) for x = -2, -1, 0, 1, 2, 3 respectively.

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A087205(n), A140165(n+1), A016116(n+1), A000045(n+2), A000079(n), A122367(n), A006012(n), A052961(n), A154626(n) for x = -3, -2, -1, 0, 1, 2, 3, 4 respectively . (END) - From Philippe Deléham, Feb 26 2012

T(n,k) = A208748(n,k)/2^k. - Philippe Deléham, Mar 05 2012

EXAMPLE

First five rows:

1

0...2

0...1...3

0...1...2...5

0...1...2...5...8

First five polynomials v(n,x):

1, 2x, x + 3x^2, x + 2x^2 + 5x^3, x + 2x^2 + 5x^3 + 8x^4.

MATHEMATICA

u[1, x_] := 1; v[1, x_] := 1; z = 13;

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

v[n_, x_] := x*u[n - 1, x] + 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[%]  (* A208342 *)

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

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

TableForm[cv]

Flatten[%]  (* A208343 *)

CROSSREFS

Cf. A208343.

Cf. A084938, A000045, A000079,

Sequence in context: A096087 A128138 A308999 * A029324 A227551 A029318

Adjacent sequences:  A208340 A208341 A208342 * A208344 A208345 A208346

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 25 2012

STATUS

approved

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Last modified October 20 08:16 EDT 2019. Contains 328253 sequences. (Running on oeis4.)