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A209416 Triangle of coefficients of polynomials v(n,x) jointly generated with A209415; see the Formula section. 3
1, 1, 2, 1, 3, 3, 1, 5, 7, 4, 1, 6, 15, 14, 5, 1, 8, 23, 36, 25, 6, 1, 9, 36, 69, 76, 41, 7, 1, 11, 48, 123, 176, 147, 63, 8, 1, 12, 66, 192, 355, 400, 266, 92, 9, 1, 14, 82, 292, 635, 910, 834, 456, 129, 10, 1, 15, 105, 410, 1065, 1833, 2131, 1626, 747, 175, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Alternating row sums:  1,-1,1,-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, 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 01 2012

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

u(n,x) = x*u(n-1,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 01 2012. (Start)

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

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

T(n,k) = 2*T(n-1,k-1) + T(n-2,k) + T(n-2,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)

EXAMPLE

First five rows:

1

1...2

1...3...3

1...5...7....4

1...6...15...14...5

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

From Philippe Deléham, Apr 01 2012: (Start)

(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, 5, 7, 4, 0

1, 6, 15, 14, 5, 0

1, 8, 23, 36, 25, 6, 0 . (End)

MATHEMATICA

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

u[n_, x_] := x*u[n - 1, 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[%]    (* A209415 *)

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

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

TableForm[cv]

Flatten[%]    (* A209416 *)

CoefficientList[CoefficientList[Series[(1 + x - 2*y*x - y*x^2 + y^2*x^2)/(1 - 2*y*x - x^2 - y*x^2 + y^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)

CROSSREFS

Cf. A209416, A208510.

Sequence in context: A100578 A061315 A144265 * A122075 A185675 A153341

Adjacent sequences:  A209413 A209414 A209415 * A209417 A209418 A209419

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 09 2012

STATUS

approved

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Last modified October 15 05:43 EDT 2019. Contains 328026 sequences. (Running on oeis4.)