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A210557 Triangle of coefficients of polynomials u(n,x) jointly generated with A210558; see the Formula section. 4
1, 1, 2, 1, 3, 5, 1, 4, 10, 12, 1, 5, 16, 30, 29, 1, 6, 23, 56, 87, 70, 1, 7, 31, 91, 185, 245, 169, 1, 8, 40, 136, 334, 584, 676, 408, 1, 9, 50, 192, 546, 1158, 1784, 1836, 985, 1, 10, 61, 260, 834, 2052, 3850, 5312, 4925, 2378, 1, 11, 73, 341, 1212, 3366 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Row sums: powers of 3 (see A000244).

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

Subtriangle of (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, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 23 2012

Up to reflection at the vertical axis, this triangle coincides with the triangle given in A164981, i.e. the numbers are the same just read row-wise in the opposite direction. [Christine Bessenrodt, Jul 20 2012]

LINKS

Table of n, a(n) for n=1..61.

FORMULA

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

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

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

Contribution from Philippe Deléham, Mar 23 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.

EXAMPLE

First five rows:

1

1...2

1...3...5

1...4...10...12

1...5...16...30...29

First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 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, 3, 5, 0

1, 4, 10, 12, 0

1, 5, 16, 30, 29, 0 . Philippe Deléham, Mar 23 2012

MATHEMATICA

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

u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;

v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;

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[%]   (* A210557 *)

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

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

TableForm[cv]

Flatten[%]   (* A210558 *)

CROSSREFS

Cf. A210558, A208510, A164981.

Sequence in context: A297595 A049069 A030237 * A118243 A210233 A297582

Adjacent sequences:  A210554 A210555 A210556 * A210558 A210559 A210560

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 22 2012

STATUS

approved

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Last modified February 19 06:46 EST 2018. Contains 299330 sequences. (Running on oeis4.)