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A208751 Triangle of coefficients of polynomials u(n,x) jointly generated with A208752; see the Formula section. 3
1, 1, 2, 1, 6, 2, 1, 12, 12, 2, 1, 20, 40, 18, 2, 1, 30, 100, 86, 24, 2, 1, 42, 210, 294, 150, 30, 2, 1, 56, 392, 812, 656, 232, 36, 2, 1, 72, 672, 1932, 2268, 1240, 332, 42, 2, 1, 90, 1080, 4116, 6624, 5172, 2100, 450, 48, 2, 1, 110, 1650, 8052, 17028, 17996 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

Subtriangle of the triangle T(n,k) given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1, 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..61.

FORMULA

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

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

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

From Philippe Deléham, Mar 17 2012: (Start)

As DELTA-triangle with 0 <= k <= n:

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

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1), 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,  6,  2;

  1, 12, 12,  2;

  1, 20, 40, 18,  2;

First five polynomials u(n,x):

  1

  1 +  2x

  1 +  6x +  2x^2

  1 + 12x + 12x^2 +  2x^3

  1 + 20x + 40x^2 + 18x^3 + 2x^4

From Philippe Deléham, Mar 17 2012: (Start)

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

  1;

  1,   0;

  1,   2,   0;

  1,   6,   2,   0;

  1,  12,  12,   2,   0;

  1,  20,  40,  18,   2,   0;

  1,  30, 100,  86,  24,   2,   0; (End)

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. A208752, A208510.

Sequence in context: A286030 A208905 A208749 * A133200 A103881 A101024

Adjacent sequences:  A208748 A208749 A208750 * A208752 A208753 A208754

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 01 2012

STATUS

approved

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Last modified October 21 20:32 EDT 2020. Contains 337925 sequences. (Running on oeis4.)