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A209127 Triangle of coefficients of polynomials v(n,x) jointly generated with A209126; see the Formula section. 3
1, 0, 2, 0, 2, 3, 0, 2, 5, 5, 0, 2, 7, 12, 8, 0, 2, 9, 21, 25, 13, 0, 2, 11, 32, 53, 50, 21, 0, 2, 13, 45, 94, 124, 96, 34, 0, 2, 15, 60, 150, 250, 273, 180, 55, 0, 2, 17, 77, 223, 445, 617, 577, 331, 89, 0, 2, 19, 96, 315, 728, 1212, 1444, 1181, 600, 144, 0, 2, 21 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

u(n,n)=(1,2,3,5,8,13,21,...)=A000045(n+1), Fibonacci numbers.

Alternating row sums: (1,-2,1,-2,1,-2,1,-2,...

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

As triangle T(n,k) with 0<=k<=n, it is (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 21 2012

LINKS

Table of n, a(n) for n=1..69.

FORMULA

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

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 21 2012

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

EXAMPLE

First five rows:

1

0...2

0...2...3

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

0...2...7...12...8

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

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*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[%]    (* A209126 *)

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

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

TableForm[cv]

Flatten[%]    (* A209127 *)

CROSSREFS

Cf. A209126, A208510.

Sequence in context: A067871 A198632 A060155 * A127954 A198061 A265583

Adjacent sequences:  A209124 A209125 A209126 * A209128 A209129 A209130

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified October 14 04:44 EDT 2019. Contains 327995 sequences. (Running on oeis4.)