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A208752 Triangle of coefficients of polynomials v(n,x) jointly generated with A208751; see the Formula section. 3
1, 2, 1, 3, 5, 1, 4, 14, 8, 1, 5, 30, 34, 11, 1, 6, 55, 104, 63, 14, 1, 7, 91, 259, 253, 101, 17, 1, 8, 140, 560, 806, 504, 148, 20, 1, 9, 204, 1092, 2178, 1966, 884, 204, 23, 1, 10, 285, 1968, 5202, 6412, 4090, 1420, 269, 26, 1, 11, 385, 3333, 11286, 18238 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

Subtriangle of the triangle given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 17 2012

LINKS

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

FORMULA

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

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

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

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1), T(1,0) = 1, T(2,0) = 2, T(2,1) = 1, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 17 2012

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

EXAMPLE

First five rows:

1

2...1

3...5....1

4...14...8....1

5...30...34...11...1

First five polynomials v(n,x):

1

1 + 2x

1 + 6x + 2x^2

1 + 12x + 12x^2 + 2x^3

1 + 20x + 40x^2 + 18x^3 + 2x^4

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

1

0, 1

0, 2, 1

0, 3, 5, 1

0, 4, 14, 8, 1

0, 5, 30, 34, 11, 1. - Philippe Deléham, Mar 17 2012

MATHEMATICA

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

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

v[n_, 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[%]    (* A208751 *)

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

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

TableForm[cv]

Flatten[%]    (* A208752 *)

CROSSREFS

Cf. A208751, A208510.

Sequence in context: A180906 A153277 A104029 * A119308 A110197 A124819

Adjacent sequences:  A208749 A208750 A208751 * A208753 A208754 A208755

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 01 2012

STATUS

approved

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Last modified October 16 07:30 EDT 2019. Contains 328051 sequences. (Running on oeis4.)