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A209135 Triangle of coefficients of polynomials u(n,x) jointly generated with A209136; see the Formula section. 3
1, 2, 1, 3, 5, 3, 4, 14, 16, 5, 5, 30, 54, 39, 11, 6, 55, 144, 171, 98, 21, 7, 91, 329, 561, 503, 229, 43, 8, 140, 672, 1534, 1928, 1380, 532, 85, 9, 204, 1260, 3690, 6106, 6084, 3636, 1203, 171, 10, 285, 2208, 8058, 16852, 21890, 18060, 9249, 2694 (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 (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 11 2012

LINKS

Table of n, a(n) for n=1..54.

FORMULA

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

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

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

From Philippe Deléham, Apr 11 2012: (Start)

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

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

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 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;

  2,  1;

  3,  5,  3;

  4, 14, 16,  5;

  5, 30, 54, 39, 11;

First three polynomials u(n,x):

  1

  2 + x

  3 + 5x + 3x^2

From Philippe Deléham, Apr 11 2012: (Start)

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

  1;

  1,  0;

  2,  1,  0;

  3,  5,  3,  0;

  4, 14, 16,  5,  0;

  5, 30, 54, 39, 11,  0; (End)

MATHEMATICA

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

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

v[n_, x_] := 2 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[%]    (* A209135 *)

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

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

TableForm[cv]

Flatten[%]    (* A209136 *)

CROSSREFS

Cf. A209136, A208510.

Sequence in context: A069931 A209152 A209158 * A258236 A258244 A258248

Adjacent sequences:  A209132 A209133 A209134 * A209136 A209137 A209138

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Mar 05 2012

STATUS

approved

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Last modified June 23 21:09 EDT 2021. Contains 345402 sequences. (Running on oeis4.)