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A209695 Triangle of coefficients of polynomials u(n,x) jointly generated with A209696; see the Formula section. 3
1, 1, 2, 1, 5, 5, 1, 8, 18, 12, 1, 11, 40, 58, 29, 1, 14, 71, 164, 175, 70, 1, 17, 111, 357, 601, 507, 169, 1, 20, 160, 664, 1550, 2048, 1428, 408, 1, 23, 218, 1112, 3346, 6106, 6632, 3940, 985, 1, 26, 285, 1728, 6394, 15012, 22442, 20680, 10701, 2378 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

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

Subtriangle of the triangle given by (1, 0, 1/2, -1/2, 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 24 2012

LINKS

Table of n, a(n) for n=1..55.

FORMULA

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

Contribution from Philippe Deléham, Mar 24 2012. (Start)

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

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

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

EXAMPLE

First five rows:

1

1...2

1...5....5

1...8....18...12

1...11...40...58...29

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

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

1

1, 0

1, 2, 0

1, 5, 5, 0

1, 8, 18, 12, 0

1, 11, 40, 58, 29, 0 . - Philippe Deléham, Mar 24 2012

MATHEMATICA

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

u[n_, x_] := x*u[n - 1, x] + (x + 1)*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[%]    (* A209695 *)

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

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

TableForm[cv]

Flatten[%]    (* A209696 *)

CROSSREFS

Cf. A209696, A208510.

Sequence in context: A304462 A021468 A209830 * A033282 A126350 A204111

Adjacent sequences:  A209692 A209693 A209694 * A209696 A209697 A209698

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 13 2012

STATUS

approved

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Last modified October 22 00:52 EDT 2019. Contains 328315 sequences. (Running on oeis4.)