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A208339 Triangle of coefficients of polynomials v(n,x) jointly generated with A208838; see the Formula section. 5
1, 1, 3, 1, 4, 7, 1, 5, 13, 17, 1, 6, 20, 40, 41, 1, 7, 28, 72, 117, 99, 1, 8, 37, 114, 241, 332, 239, 1, 9, 47, 167, 425, 769, 921, 577, 1, 10, 58, 232, 682, 1492, 2368, 2512, 1393, 1, 11, 70, 310, 1026, 2598, 5008, 7096, 6761, 3363, 1, 12, 83, 402, 1472 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Subtriangle of the triangle given by (1, 0, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 27 2012

LINKS

Table of n, a(n) for n=1..60.

FORMULA

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

v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x),

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

Contribution from Philippe Deléham, Mar 27 2012: (Start)

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

G.f.: (1-2*y*x+2*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(2,1) = 3, T(1,1) = T(2,2) = 0 nd T(n,k) = 0 if k<0 or if k>n. (End)

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

EXAMPLE

First five rows:

1

1...3

1...4...7

1...5...13...17

1...6...20...40...41

First five polynomials v(n,x):

1

1 + 3x

1 + 4x + 7x^2

1 + 5x + 13x^2 + 17x^3

1 + 6x + 20x^2 + 40x^3 + 41x^4

Contribution from Philippe Deléham, Mar 27 2012: (Start)

(1, 0, -2/3, 2/3, 0, 0,...) DELTA (0, 3, -2/3, -1/3, 0, 0,...) begins :

1

1, 0

1, 3, 0

1, 4, 7, 0

1, 5, 13, 17, 0

1, 6, 20, 40, 41, 0. (End)

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] + 2 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[%]   (* A208338 *)

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

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

TableForm[cv]

Flatten[%]   (* A208339 *)

CROSSREFS

Cf. A208338.

Sequence in context: A086273 A054143 A104746 * A185722 A287376 A209418

Adjacent sequences:  A208336 A208337 A208338 * A208340 A208341 A208342

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Feb 27 2012

STATUS

approved

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Last modified October 15 04:33 EDT 2019. Contains 328026 sequences. (Running on oeis4.)