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A208337 Triangle of coefficients of polynomials v(n,x) jointly generated with A208836; see the Formula section. 5
1, 1, 2, 1, 3, 3, 1, 4, 7, 5, 1, 5, 12, 15, 8, 1, 6, 18, 31, 30, 13, 1, 7, 25, 54, 73, 58, 21, 1, 8, 33, 85, 145, 162, 109, 34, 1, 9, 42, 125, 255, 361, 344, 201, 55, 1, 10, 52, 175, 413, 701, 850, 707, 365, 89, 1, 11, 63, 236, 630, 1239, 1806, 1918, 1416, 655 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

coef. of x(n-1) in u(n,x): A000045(n), Fibonacci numbers

coef. of x(n-1) in v(n,x): A000045(n+1)

row sums, u(n,1):  A000129

row sums, v(n,1):  A001333

alternating row sums, u(n,-1): 1,0,1,0,1,0,1,0,1,0,...

alternating row sums, v(n,-1): 1,-1,1,-1,1,-1,1,-1,...

Subtriangle of the triangle given by (1, 0, -1/2, 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, Apr 09 2012

LINKS

Table of n, a(n) for n=1..65.

C. Kimberling, Enumeration of paths, compositions of integers and Fibonacci numbers, Fib. Quarterly 39 (5) (2001) 430-435 Figure 2.

FORMULA

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

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

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

From Philippe Deléham, Apr 09 2012: (Start)

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

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

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

G.f.: -(1+x*y)*x*y/(-1+x*y+x^2*y^2+x). - R. J. Mathar, Aug 11 2015

EXAMPLE

First five rows:

  1

  1...2

  1...3...3

  1...4...7....5

  1...5...12...15...8

First five polynomials v(n,x):

  1

  1 + 2x

  1 + 3x + 3x^2

  1 + 4x + 7x^2 + 5x^3

  1 + 5x + 12x^2 + 15x^3 + 8x^4

(1, 0, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, ...) begins :

  1

  1, 0

  1, 2, 0

  1, 3, 3, 0

  1, 4, 7, 5, 0

  1, 5, 12, 15, 8, 0

  1, 6, 18, 31, 30, 13, 0

  1, 7, 25, 54, 73, 58, 21, 0 . Philippe Deléham, Apr 09 2012

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 + 1)*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[%]    (* A208336 *)

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

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

TableForm[cv]

Flatten[%]    (* A208337 *)

Table[u[n, x] /. x -> 1, {n, 1, z}] (*u row sums*)

Table[v[n, x] /. x -> 1, {n, 1, z}] (*v row sums*)

Table[u[n, x] /. x -> -1, {n, 1, z}](*u alt. row sums*)

Table[v[n, x] /. x -> -1, {n, 1, z}](*v alt. row sums*)

CROSSREFS

Cf. A208336.

Sequence in context: A055129 A133804 A185943 * A208335 A208597 A179943

Adjacent sequences:  A208334 A208335 A208336 * A208338 A208339 A208340

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 26 2012

STATUS

approved

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Last modified October 15 12:31 EDT 2019. Contains 328026 sequences. (Running on oeis4.)